0:12

When you solve these problems, this is really how I want you.

Â It's kind of a cookbook formula.

Â Those in 3200 have already seen this a little bit, but

Â it's how do you approach kinematics.

Â This is purely kinematics.

Â I'm not throwing in forces, some torques, some mass.

Â I'm just saying I have a position vector or some vectorial quantity.

Â I want to take its first derivative.

Â I want to take its second derivative.

Â But I've written things in terms of rotating frames.

Â How do I do this, right?

Â 0:45

Just looking at planar motion.

Â So here's an E frame with e1, e2 and

Â then e3 is pointing out of the board, right?

Â And this is a vector r.

Â And this is a particle P that I'm tracking.

Â So I'm giving you actually quite a bit of information here.

Â What I want to do is, the question is what is the natural derivative of r?

Â Let me be explicit.

Â This is the point O, the origin.

Â Let me get rid of E.

Â That's O.

Â 1:26

You know what?

Â Let me get rid of this as well.

Â I'll define this separately.

Â So you have this given.

Â Some frame.

Â Some vector.

Â It's planar motion that we're looking at and

Â it'll make the math a little bit easier to do it here quickly.

Â And we can go through this.

Â So the first step that you have to do is write vector.

Â Whatâ€™s the vector?

Â If I could write nicely, that we are looking for?

Â And in this case, we need the motion of P relative to this point O.

Â If you want to be explicit, you could write P relative to O, but

Â I'm going to use an r shorthand just so

Â I don't have a lot of P relative to Os floating around, all right?

Â That's what I need.

Â What's the laziest way?

Â Laziness is a virtue here.

Â What's the easiest, laziest way that we can write this r vector to go from O to P?

Â 2:58

Let's just call it L.

Â We're just making up this problem.

Â There's a certain distance, d or L.

Â Now what did you call the other one, Chuck?

Â >> [INAUDIBLE] >> Okay,

Â not what I would've called it, but that's good.

Â That makes it live and real.

Â 3:18

So you're always trying to trick me.

Â r hat, so r hat has to be unit direction vector, but it's basically saying,

Â hey, that point P is 4 meters in that direction, that's it.

Â You've got a question?

Â No, okay, just you're raising your hand.

Â [LAUGH] That's probably the easiest way.

Â But now r hats, if we want to use this, r hat is going to be a rotating vector.

Â 4:14

Get all omegas basically.

Â But to get the omegas, we have to have full frame definitions.

Â Earlier we're talking about, could we have a frame around the velocity vector, and

Â that's not really possible.

Â You can lock in one direction but the rotation about that,

Â there's an infinity of possible frames that could do it, right?

Â So we have to, here too, I only have one vector.

Â So you're going to have to define two other vectors to fully set up this frame.

Â 4:47

Need to be orthogonal, we know that right-handed, right,

Â unit length, all this kind of stuff.

Â Jordan, what do you think?

Â >> Direction out the board, the same for both frames.

Â >> Okay, so here you want to,

Â now I'm going to define e3 is out of the board, all right?

Â And I need a vector here out of the board.

Â Good, so we can share e3 between two frames.

Â Where is the third vector going to go?

Â [INAUDIBLE] It's r.

Â >> Up? >> That way.

Â >> Could it also go down?

Â >> Yep, unless you had three were into the board.

Â >> No, okay, I'm glad you said that.

Â Let's draw what I agree.

Â This is the easiest way for me typically.

Â What do we want to call this vector?

Â 5:37

See, I would've had a frame P, with p1, p2, p3, which would've been much easier.

Â But no, you guys have to get complicated on me.

Â Okay, so here we go.

Â We have r hat, theta hat, and e3.

Â I told you you could mix names, and you're proving me right.

Â Okay, so we can do this.

Â So we have to define the frames.

Â 6:01

And how many frames do you need is really directly related to how many omegas you're

Â going to need.

Â If you have all the different vectors that you need to get from A to B,

Â B to C, D to E, you may need many frames.

Â And then for, you'll need many omegas.

Â In this case, it's all planar motion.

Â I agree with Jordan, you could have the third one skewed, but why?

Â Then you have all these weird, orthogonal angles to do.

Â So this would work.

Â So let's define this frame.

Â Point P.

Â What is the first axis now, Jordan?

Â >> r hat.

Â >> r hat, theta hat and e3.

Â You really should write down these frames.

Â Because the ordering is going to be important, especially when you guys get

Â creative as you are, not just doing p1, 2, 3, as I would have done.

Â But you are.

Â We're going to do r hat, theta hat, and e3.

Â Now I'm going to go back quickly to Jordan's earlier comment.

Â He said it had to be up in this direction.

Â Is there any way I could have had theta hat point down here and

Â still define this P frame to be right-handed?

Â 7:07

>> [INAUDIBLE] >> No, we're keeping e3,

Â we're not touching e3, we're not touching r hat.

Â I'm only flipping theta hat.

Â What must change in this definition to make it right-handed?

Â She used the right hand, right-handed, not left left-handed.

Â Sorry? >> [INAUDIBLE]

Â >> If you would flip the definition and

Â say my first vector is theta hat, in which case that would be here,

Â my second vector is r, then the third one is out of the board.

Â So you could actually do it and make it right-handed but

Â now you're really making things confusing.

Â [LAUGH] So whatever is easiest.

Â It all gets you to the right answer.

Â It's just names, and it's good, in the problems, to mix it up.

Â because I shouldn't confuse you because instead of point body frame B,

Â I made body frame T.

Â And now all hell broke loose, you know?

Â And then that means you didn't understand the stuff,

Â you were just plugging in formulas.

Â Matt? >> [INAUDIBLE]

Â 8:00

It also has an order when you put them

Â into matrix form.

Â because then in the P frame, I understand the first scalar in that prebound one

Â is times the first vector direction.

Â And then it matters as well.

Â So let me do tensors and so forth, it did all, they didn't matters.

Â >> [INAUDIBLE] to the problem?

Â >> No.

Â Laziness is my convention and it typically gets where we come from what's here

Â I would use r1, that's my first one and then I build everything around it and

Â that tends to make my life easier.

Â But there's no real convention about this.

Â So good, so write the vectors.

Â We get from here to here.

Â The easiest way typically is from here to here and I just define a frame that goes,

Â well I need to go two meters that a way.

Â And that away r hat, right?

Â And then we need to flesh it out, Jordan called the other directions theta hat and

Â e3 hat, and the rest gets there.

Â Now we need omegas.

Â 9:24

>> And so omega is theta dot where theta is the angle between r and e one and

Â with an arrow in the upwards direction.

Â >> Thank you, that was my next question.

Â Right.

Â You put your thumb along e three,

Â curl your fingers, that would be a positive angular rotation, perfect,

Â we got theta now that's so we need theta dot and what's the axis?

Â >> Three.

Â >> D 3 that's it.

Â When you think of angular velocities as simply magnitude times the direction.

Â That's how I break them and down two.

Â I go okay this frame and this frame, what's going on?

Â Well, they're rotating about this axis and in these problems we're solving right now,

Â they're often rotating about some common axis.

Â Here we're rotating both frames about E3 and that makes it a lot simpler.

Â If it's a 4D, 3D tumble, you will see in chapter three how we handle those omegas.

Â It's just a little bit more bookkeeping.

Â What we need, right?

Â So good. So let's write this out.

Â We've written position, gotten frames, and now we differentiate.

Â 11:02

>> What defines a frame to be inertial.

Â >> Non accelerating >> Non accelerating.

Â If you're rotating, there's centrifugal accelerations immediately .If

Â something's rotating default boom, not inertial, right?

Â But if you're accelerator and going faster and faster and faster,

Â you are not an inertial frame, right?

Â Could you be traveling at a constant speed?

Â 11:29

Yes.

Â because in that case, there's zero acceleration.

Â So you don't have to be a station.

Â Some people say inertial frame means stationary frame.

Â If you say that in the prelim, I'm going to raise all kinds of flags and

Â say, wait a minute.

Â Give me more details.

Â It doesn't have to be stationary,

Â it just has to be non-accelerating, that's what it boils down to.

Â So hopefully it's something you've already heard of.

Â So now, we're looking for r dot, that's defined to be the, I'll make it explicit.

Â 11:57

Would you like to differentiate this directly as seen by an end frame.

Â Hopefully the answer is no, because out hat doesn't.

Â It varies the time, it's going to be none zero.

Â We'll have to figure this things out.

Â Maybe take vector components, I will introduce all of the signs and co-signs,

Â and I'm way too lazy to do that, okay?

Â So, as seen by what frame is the derivative of this right hand side,

Â could be really easy.

Â 13:29

>> R Hat. >> R Hat.

Â That's why.

Â Right?

Â Because immediately, with respect to what frame or

Â all these crazy rotating parts not going to matter.

Â That's essentially what we're doing with the transport theorem.

Â You're always choosing a frame where it's almost like inertial.

Â It just means hey, I can just treat this vector stuff as fixed things and

Â not worry about them.

Â That's the trick.

Â That's the essence of the transport theorem.

Â So that means if I write this out,

Â I have a ddt(r) times r hat and it's being very explicit right now.

Â You could do. You could skip a lot of these steps and

Â put them together.

Â Times DDT of R hat there.

Â [INAUDIBLE] by the P frame.

Â So this whole thing is.

Â Shit. I shouldn't have.

Â I messed it up.

Â 14:21

So the p derivative is going to be this, right?

Â Then should I write a p over here on the dt of R?

Â >> [INAUDIBLE] >> You can.

Â >> [INAUDIBLE] >> So

Â what is a p frame derivative of a scalar?

Â [INAUDIBLE] I

Â would say if you're doing a derivative of a scalar, it's just a time derivative.

Â Don't put frames in there, because I've seen too many people,

Â especially this homework, this is one problem,

Â I think it's 3.6 that you'll be going through, that's really fun.

Â Where all these little subtleties matter and all of a sudden people put transport

Â theorems on scalars and have omegas cross the scalars and

Â doing all kinds of crazy stuff that makes absolutely no sense.

Â So if you just have a scalar you just doing a time derivative.

Â I would say just write just a time derivative, that's way more rigorous.

Â I agree, if this is composed of vectors and

Â to vectors you could take derivative c anything.

Â 15:24

But this is just a time derivative, so here we said this is going to go to 0,

Â so you only really just have r.

Â R hat. And r is just a scalers such as a time

Â derivative of that times r hat.

Â Now to get this derivative,

Â 15:59

Sorry, yep, right there, Matt, thank you.

Â >> [INAUDIBLE] >> Which is?

Â >> [INAUDIBLE] >> Good, so

Â that was my next question; which letters go here?

Â So your saying q with respect to E, no, P.

Â Must be P because we have a P-frame, crossed with r that we do there, yep.

Â because we took the P-frame derivative here, so we need omega P relative to E.

Â Again, that thing's just placeholders with the letters.

Â Yes, Marion?

Â >> [INAUDIBLE]

Â >> We can, give me a better name?

Â >> [INAUDIBLE] >> [LAUGH]

Â >> Better is relative.

Â >> [LAUGH] >> But we can, okay, so so far,

Â we didnt need it yet.

Â So we'll make it a q-hat, and we'll make this a q.

Â 17:00

Absolutely, so now we have to compute this.

Â So the first part is r-dot r-hat +

Â omega is theta-dot E3 x with r, which is r r-hat.

Â So the first part is still the same, right?

Â Now this crossed product, the scalars you can bunch up together.

Â And just have r theta-dot, what is E3 x r?

Â Now with these names, see, it's the 3rd vector crossed with

Â the 1st gives you 2nd, right, and plus the 2nd.

Â Once you have the right term, make sure it's plus and minus.

Â In this case, it's plus.

Â So in this case, the newly-christened q-hat appears.

Â Again, I would've used P1, 2, and 3, but that's just me.

Â 17:52

And that's it, that is our inertial derivative of this.

Â And you can see, we've solved it in a very quoted frame agnostic way.

Â We're just saying, you have to somehow know r-hat is and q-hat is, and

Â they're orthogonal.

Â And so if you have this frame defined in this problem, this is the theta angle,

Â now you can do your sines and cosines.

Â And put it in MATLAB, and compute an actual matrix representation in

Â the n-frame, the b-frame, whatever frame you want.

Â But we're only doing the frames when we need to, at the very, very end.

Â 18:26

It does not mean put every component of this vector into the inertial frame.

Â If anything, you're really going to aggravate me.

Â I want you, in the homework, to use rotating frames.

Â If I see lots of sines and cosines,

Â I'm probably just going to get my red pen out and slashing off points.

Â Use rotating frames, that's the point.

Â These are really simple, boring problems, all right?

Â The purpose here is practice how to use rotating frames.

Â Follow these steps, get through this stuff, and

Â then you can come up with some way to write it.

Â In some of the problems, there's ambiguities.

Â I don't always give you the e-frames.

Â There's offsets you can put in, so

Â everybody's answers might be slightly different.

Â And that's perfectly fine, all right?

Â But this is the steps.

Â We only assign corner frames when we absolutely have to.

Â And that's probably at the very end, if you want to get actual numerical answers

Â to compute something than using all these states, okay?

Â Any questions about this?

Â 19:23

Let's just set up some other problems that relate a little bit to homework.

Â So in this case, this was simple.

Â If you did have to get inertial acceleration,

Â let's just talk through that.

Â So we have our dot, All right?

Â And the problem statement part B says give me the inertial acceleration of this.

Â What are the steps that we have to do in this case?

Â 19:59

Yes, go ahead, Kyle, right?

Â >> [INAUDIBLE] >> Casey, okay, I was off, Casey.

Â >> [INAUDIBLE] >> What frame would

Â you differentiate this to make life easiest?

Â >> [INAUDIBLE] >> The P-frame that we had, right?

Â because then r-hat, q-hat are all fixed, right, when you do this, +.

Â Which omega, Casey, do you need here?

Â >> [INAUDIBLE] >> E in this case.

Â >> [INAUDIBLE] >> E was the n-one,

Â crossed with the vector, itself, and you carry it out, all right?

Â So let's look at different problems.

Â I'm just going to quickly show you some,

Â we got two minutes to show quick highlights and try to mix things up.

Â Let's say we have a position vector that is a a1-hat because it's a frame,

Â a 1, 2, and 3.

Â Somebody was nicely lazy.

Â And b b1-hat, there's a b-frame, b1, 2, and 3.

Â Now I have to get a derivative of this.

Â As seen by what frame would you choose to differentiate this?

Â 21:15

Differentiation is a linear operator, right?

Â So what does that mean?

Â How can you solve this?

Â How can you be extra lazy and make this really easy?

Â >> [INAUDIBLE] >> Yes, exactly, so

Â if your vector has components in this frame and

Â components in that frame, pick all the ones that are in one frame.

Â So I would say this part is going to be a A-frame derivative +.

Â Which omega do we need to use here?

Â 21:44

because I need to find an inertial derivative in the end, so

Â what omega would go here?

Â >> A with respect to n.

Â >> A with respect to n crossed with the vector itself.

Â And then this part would become a B-frame derivative, all right,

Â of this stuff + omega B with respect to n crossed with this stuff again, right?

Â So just remember that because with that,

Â no matter how complicated this stuff looks, you can always, this is all the q,

Â this is all the p, this is all the s, chunk them together.

Â But you have to have figured out the proper omegas.

Â And that's again, note, omega's a vector.

Â So if you have omega an,

Â and omega ba, you could add them to get omega bn, or something.

Â You have to do the proper vector math to find these things.

Â So that's good, one last problem, yeah.

Â What if, Let's just put us into orbit.

Â Here's the spacecraft, and this is the orbit frame.

Â 22:46

Here that's radial, tangential, and orbit normal, ih, all right?

Â That's there, so that's an orbit frame,

Â defined this way, {ir, i theta, ih}.

Â One homework problem in particular deals with this.

Â And that is you have the position vector here, r.

Â 23:47

Do we take an N derivative, Andrew?

Â >> [INAUDIBLE] >> Which one?

Â >> [INAUDIBLE] >> Yeah, space station one, right?

Â So check for that.

Â Most of the problems always ask for inertial, inertial, inertial derivative,

Â inertial velocity, inertial acceleration.

Â But there's some that, all of a sudden, things are twisting and rotating and

Â you're on this Ferris wheel or something.

Â And it says, hey,

Â how does this vectorial quantity change as seen by an observer in this other frame.

Â In that case, you're picking an O-frame.

Â Transport theorem still applies, you just do the same stuff.

Â Yes, Matt?

Â >> [INAUDIBLE] >> Yes,

Â you may have to flip the, >> [INAUDIBLE] You have

Â the rotation rates relative to [CROSSTALK] >> Yeah, because you may have this and

Â say, look, I can easily take the derivative in N for some reason.

Â Maybe this is given in N-frame stuff.

Â But then you would need omega N relative to O crossed with that

Â just to complete the transport theorem.

Â Again, these letter are perfectly interchangeable.

Â Just look out for what the problem statement says.

Â So if it's asking for

Â inertial derivative or a-frame derivative, it's just how you differentiate it.

Â It's differentiated as seen by these observers.

Â You can use mixed frames.

Â Doesn't have to be all in the e-frame or the b-frame.

Â You can mix the frames unless, if I need specifics,

Â and I don't think any of these homeworks ask for it, I would say,

Â hey, express your answer in terms of e-frame components.

Â That means at some point you have to do your sines and cosines and

Â map everything into one frame.

Â But I don't typically here.

Â I'm really trying to encourage you to use rotating frames.

Â That's the whole purpose of this, okay?

Â We're a few minutes over, but that's good.

Â We'll pick up here Tuesday.

Â Start these homeworks, come back with good questions.

Â Let's see where you get stuck.

Â As you apply it, that's always when all the little intricacies come in.

Â