So next we're going to talk about continuous systems. That's kind of next thing. We're moving away from kingdom attics, you know that now we've reviewed it quickly but intensely. So we're going to start out with jello, which is kind of nice, in 5010, this is a stepping stone because we really just do everything rigid. It's primarily a rigid body, a little bit multi rigid body with the spacecraft and a wheel or a dual spinner. In this class, we're going to go back into what if it does change shape? What if it does deform? Especially when we go to analytic mechanics. So when we go to the other parts. So this is a good review of just the basic concept again, and just get depending on your background, everybody back on the same notation. So, if we're talking about jellos in space, what are we talking about here? I've drawn something. Could you give me an example that does not satisfy this blob? So let's do example by exclusion. Could this be a spacecraft? Yes. Could this be space station? Could this be the space station with fuel on board? Liquid fuel on board? Yeah. Because it could change shape and doing that. Could this be the space station with liquid fuel while you're thrusting and expelling fuel? Why not draw? So there's these things called closed and open dynamical systems and we're still talking about closed dynamical systems. So at some point. So if we have multiple wheels or multiple spacecraft and their docking and robotic defectors and they're grabbing it still, you can somehow draw a dashed line around this whole thing and saying that's my dynamical system. It's the doct thing. It's the fuel. But if you have fuel that you're expelling and just ignoring, then that's what's called an open dynamical systems, not a closed and some of these things don't apply. So just a little bit of a refinement. Now, we're definitely talking about closed dynamical systems. This could be jello, this could be fuel your modeling, this could be a spacecraft. This could be a deployable spacecraft with structures like panels like the MARS mission that cannot fold it out one way and then another, that was a maneuver with variable shape and what happens there. So Newton's law still applies. You've written it out here in differential form. Later on, we're going to go back to this again when we go to the very fundamentals of particles and anything becomes a system of particles. And use these, Lagrange and Donald bears all these systems. So we'll see this over and over again. And mass times r..df. So what is, dn, that's just a differential mass of whatever little particle you're looking at right differential element. These could be individual particles. Sometimes we do that, your spacecraft system might be six spacecraft in a constellation and you're looking at momentum of the full thing and you might just treat each other particle because they've given the distances, that's maybe a good approximation. Or it might be a particle that's part of a bolt or a panel or a trust of a space station and then you just have lots of them. Then it becomes a big summation. So what is the r double dot in here? Andrew. What do you think? So, I've just written big r double dot. So how is that defined to be inertial acceleration? What must you do with our, what kind of dots are you taking? No, how do we have to define our to find an inertial acceleration are points to the mass element. But what's the origin of the vector? Could you point, could you attach that origin to any frame Lindsay? What do you think? Yeah, what does the inertial frame mean? Not accelerating. So you could be constantly moving. The velocities will be different, some might be moving at 10 meters per second. Another inertial frame is on the train that's moving at zero because that cup of coffee is just always ahead of you. But compared to you guys, it's moving at 10 centimeters per second. So the velocities might be different. But the accelerations as constant, velocities will drop out because they're initially constant. So that we just needed to be there. So all our equations of motion are based on you've taken your position of actors with respect to an inertial frame and then we do two dots, what are these dots again in this class time derivatives but more specific inertial. So as seen by an inertial observer, does it have to be the same as the end frame? No actually, you might have different inertial frames. It's very possible, you know, depending on how you have to find it. There's some homework problems you did with something rolling on a conveyor belt but the conveyor belt was moving at a constant speed. So you may have chosen to put something on that conveyor belt that's fine. It just has to be as seen by an inertial observer. Good, left hand side dF looks so simple what goes into that dF, Ricardo. No, well you see it defined below dFe plus dFi. Why is not correct? You said only the external forces? Yes. Okay, so the answer was that you said it has to be all of the forces, not just external forces. And the reason is we're just dealing with that bolt, that bolt is attached to a panel, there will be a force between bolt and panel that will actually keep these things in lined and the second half when we talk about holla gnomic non gnomic constraints. That's exactly what we those things are actually constraint forces that are making sure that bolt doesn't come loose on. Otherwise that constraint is not true and you have problems panels flying off, that kind of stuff. So we both have internal forces right now and we will have external forces. So what examples of external forces disappear? Yeah, gravity. That's the big one. Besides gravity. Abby, what else? Yeah, atmospheric drag as example solar radiation pressure, maybe multiple gravitational accelerations. Maybe thrusters acting on it. Maybe somebody talking if you're doing rendezvous servicing all those kind of things. Space debris, popular topic these days. Space debris, micro meteorites hitting you. You're landing on an asteroid. Just might be landing dynamics as one strong hit. All of those things are on there. And it may affect some elements and not others. This is where it gets more complicated. So good. So this is a very fundamental system but it's still a closed dynamical system that we're looking at here. Good. So what are some highlights? Ricardo was already jumping ahead. Talking about what about the full system. And so some of the key things we can look at is what? Okay, this is our dynamical system. This blob in space and each one has some forces acting on it. And if you integrate all the forces acting on every little element of the system, that's what we're doing here with this integral b. This could be a summation if it's discreet forces or just a continuum If it's a jello fuel, you know, something acting through it? It's only the summation of the external forces? What happens in the internal forces? >> The cancel on because you know what the force of the act panel is the same as the. >> Equal in opposite in direction. So whatever you're pushing against the wall, the wall is also gets pushing against you and you and the wall are both part of the dynamical systems. So for the summation, all the internal ones have to cancel out. Alright, so that's cool. So that's really it. So f now is the total system. So this is nice center of mass properties really. The center of mass is the mass average location of this dynamical system. So, I've got a 10 kg object here and a 10 kg object here. The center of mass will be right in the middle. I've got a 10 kg object to my left, A one kg object to my right. The center of mass will be somewhere over here. It's the mass average location. And different ways to write it. You can integrate our time's DM. And then you divide by M big M, the total mass. And that gives you the center of mass. The other one that we also do. What's the other center of mass definition? It's not on the slide then that we often use. So here I'm using big R and R is written relative to some inertial, non accelerating frame. What if I used little r, Gavin, do you remember? Yeah. So if you do the body integral of that and I'll just I forget somewhere. I have it on a slide. So if you do the body integral of little DM. That said okay, I had ten here. One on my right hand side. Them as average locations could be roughly here. If you take your position here. Plus this position times that mass. This one is going to be negative. This one's positive. The two of them have to average out perfectly to where the mass average location with respect to the center of mass must be zero. It must be the center of mass essentially. So this one becomes zero. These are equivalent. And then 5010, you went through that if you kind of rusty on that, it's like a two liner, go back and go through it again. Okay, cool. Moving on, center of mass. Now, if we have the center of mass property, we're going to derive the super particle theorem. So, before I show you the rest of the math, just inwards, who can Julian? Can you tell me what the super particle theorem was? Yeah, promotion of central. So the super particle theory. Another way to paraphrase you said the only external forces change the center of mass of the system. Another way to say that is you can treat the center of mass as a discrete particle and just apply ethical dilemma. This is one reason why we can do, you know, get the net force on a spacecraft and then propagate, even though it's a three dimensional object, its center of mass will act like a particle. You just have to get the right forces. And there are some approximation into forces because we assume it's a point mass. If you did a 3D mass, how would that modify the gravity? I'm jumping way ahead of what we did in 5010. But let's see if you guys can put into connections. Why is this an approximation that the Earth Is that Lindsay Lindsay? Yeah, the gravity gradients actually remember we actually did so for twerking first auto gravity gradient was important. It gives you a force that can restore or destabilize depending on your orientation. There was stable and unstable equilibrium is in motion. But there also was an expansion for this, the gravity force there, it wasn't a first order term, it was like way higher, right? And we typically found that attitude does impact your orbit. But it's like 10, 11 orders of magnitude smaller than the main stuff. And so almost anything else matters more. So it's a very good approximation. But but yeah, this is why we can treat for complicated systems. You've got panels deploying asymmetrically and fuel sloshing and you know, all this kind of weird stuff happening. But you can track the center of mass and make sure that unless there's an external force being modeled in your simulation, that center of mass better act inertial. So it has If your initial conditions are such that you have some net momentum, that momentum will carry you forth without changing, you know, So that's a great integration check when you're running simulations. If you've got multiple things happening, you know, spacecraft multiple wheels, you can just have it hover in space, not in an orbit and if it's at rest to begin with and then you have theta angles deploying panels or wheels spinning up. That's what you'll be doing in homework too. Check your center of mass. Does it stay steady, will it stay steady, Anthony? What do you think? If you're looking at the whole system perfectly? Yeah, it should. Even with like, fuel slash your pyramid slasher? It should. But in a numerical simulation, will it? Exactly. So that's always the question that people come to be in office hours. Well, this is 10 to the -8, is that small enough? Don't ask me this is your assignment, right? You're going to have to figure out in research just like well, is that small. 10 to the minus eight might be huge compared to what you're doing or it might be itsy bitsy tiny compared to what you're doing. That's part of the challenge when you use these methods to check your system, you're going to have to look at what's the net momentum. If that's 10 to the minus seven and you have 10 to the -8 variations, that's pretty large actually, something systematic might be going on. How can you check in your code if this is a numerical issue? I mean what is a easy way to check if it's a numerical issue or a systematic error in your code or logic? Sorry, step size, great refinement, basically. That's an easy one. Because should I use a one second time step, do I have to do 1/10 of a second? Do I have to go to 1/100 of a second? Well, that depends. And when you do homework too, this will be really important because you have wheels, some of them, if you go put in crazy gains, you're going to be spinning at amazing speeds. Well, that requires really is a very stiff dynamical system. All of a sudden things go up and you have to have really small time steps or a very high order integrator to do this. So you can refine your time steps and go, just took two seconds to compute what if I have my time step will make it four times smaller and you get pretty much the same error, then that error you're seeing is not because of integration, actually part of the system. If you have 10 to the minus eight and you have your time step and you get like 10 to the minus nine-ish, and you have it again, then you get 10 to the -10, then I would say, you're arguing pretty well. This is related to integration. So the integration error is always going to be part of your life. When you do numerical simulations of the analysis says it should be perfect, but your simulation is not perfect. It's a model therefore it's wrong, right. It'll have discrete assumptions where the reality that we're studying is continuous. They're just little things to be aware of. So if we do this, we have a equation MR c equal to integral Rdm. Then the arrow down shows I just took two dots over everything. I've taken to inertial derivatives of both the left and right hand side. I will have mass times the inertial acceleration of that, which must be here. This but our double dot dm back to the very first equation. That must be the total force acting on this system. And then I do the body integral of the total force and that's where the internal forces will cancel. I And in the end you have this little equation. So every closed dynamical system will act its center of mass acts just like a particle. And that's something you can check. So this is a great integration. People often check energy momentum work energy principles but center of mass is overlooked sometimes and that's a good one to check that is it? To within numerical precision, behaving the way you expect. Or if you do apply a force and this makes a panel collapse and that pushes and fuel slashes and it gets very complicated. But you can predict with this equation. This is how the center of mass should vary because I have this force one newton pushing for 10 seconds, and this is how the center of master do now what the rest of the shape does. Who knows? But it has to add up to give me exactly that again. So it's actually a really nice sanity check knowing this principle. It's not just for analysis.
