 Nu, ek heb het dit geëxciteerd, en dat is intermediaat-referent frame. Het maakt het de start van soveel dynamische, klassieke mechanische korsen en het is iets wat je moet onderstaan en het is eigenlijk, het is beautiful mathematics en het is, het is, het is, het is a wonderful thing om te kijken at en om te bedrijven en onderstaan voor de eerste keer. Echt ekciteerste stof. Dus wat gaan we doen met jy? Je gaat dit soort problemen hebben. Je gaat een cartesium coördinat systeem hebben, wat sommige mensen die het called the world view, het gaat om wat statieke spot te doen. In het mei-klaas zie je, het klasroom wordt gebruikt als het xy coördinat systeem en dan een frisbee rotating, gering door het air en een bug, op het frisbee, of wat persoon, op wat alian space ship, while the space ship is turning around and flying through the air. Wat we essentially dealing with are some others as well in a merry go round that goes around on an arm that also goes around so one of those playground things or something that turns on an arm that also turns so all of them have angular velocity. What it boils down to is that you have an inertial reference frame with standing stall and another moving reference frame. On this moving reference frame which might be purely translating so it's just moving in a straight line it might be going in a curved path it might be rotating, rotating is the famous one that's the one we're dealing with and on that body we draw a new coördinat systeem a new, now we're just dealing on a blackboard here it's a two-dimensional space on the blackboard two-dimensional space there's another body coördinat systeem it's a little x and y axis and it is rotating but imagine you know these GoPro's or drift cameras or these action cameras if they mounted on the helmet you see this quite still in the whole world is moving about you and it looks quite odd, the same thing there if this was rotating about this point about this point which I've labelled I here and if you sat and just looked in that direction and this thing is spinning it will look as if the whole world is going around and all you're seeing is straight ahead so there's a difference between this coördinat systeem and that if you're on this one any point there, a static point there is going to look completely still of course if something moves on there you're going to see it move this person here is not only going to see that movement but this whole movement as well so there's movement on top of other movement and that's what this is all about so this is a body coördinat systeem on a rigid body that's one referral other books refer to it as an intermediate reference frame an intermediate frame so here's our one reference frame you know if we work in python we're going to call this one reference frame and we're going to call this quite a different reference frame and what we want to know is the following this is the origin of this reference frame and there's this moving point P that moves in a also in a circle it has a little path that it traces we're interested in what happens over time so it's always parametrized and if something is parametrized we can express it as a vector so it's going to move in a little path this thing is rotating and moving in its own little path so I have this coördinat systeem but I want to express this movement as if I'm the person sitting here I'm the person sitting there and I've got to work out these things now first one of the questions that people ask all the time is this point fixed is this the center of mass no it's something that's happening instantaneously we taking a snapshot in time remember that's what derivatives is all about it is the limit as delta t goes to zero so in that split second that is our our zero point there our origin of our coördinat systeem in that very point time and place that point is there so how can we describe that point so I'm going to use this notation this is my point O my world view for W so W for my world view this is my intermediate frames point O which I'm calling my I for intermediate frame and this is my point P so I can say that in that split instant this vector OP is the point P in the world view so the position vector of point P in the world view point P with respect to the world view is the sum of these two vectors this vector plus that vector and what are they that is this position vector from point I so point I as seen in the world view point I as seen in the world view plus this little vector point P in the intermediate view so plus point P in the intermediate view so that's how we are going to set things up what I'm interested in is velocity so the first derivative of that and then the second derivative of that acceleration you've all seen that you can memorize those equations and that understanding it is beautiful two things that we've got to understand to do this because we're going to take derivatives first and second derivative of one reference frame within another reference frame and of course this reference frame is not standing still so when you take its derivative something special is going to happen just imagine constant circular motion because as I say this little reference frame at any one time it is moving so this I hat small lower case I hat it's going to move now just imagine in that split second I can see that little bit of movement little bit of a rotation as being part of uniform circular motion there with some angular velocity there omega and remember if that is omega it's the spinning in this anticlockwise direction the vectors pointing out to you so this is just omega k if this is my position vector and it sweeps around and around at an angular velocity it's derivative of the velocity vector is always going to be perpendicular it's always going to be perpendicular so the velocity vector is perpendicular to the position vector and because I'm seeing things in split seconds of time it is just that split second no matter what the real path is remember we looked at curvature before we looked at so it's always part of that radius of curvature so it's always on part of a circle so imagine this so I'm going to call this if I forget upper case capital I hat upper or capital J hat those are my unit vectors in those two directions in lower case smaller i smaller j that is in my intermediate frame those are my unit vectors so they have distance of one or I should say magnitude of one and they have a direction so I've blown them up here there's I hat and there's J hat and at any time they are going at an angular velocity or turning at an angular velocity of omega because we're seeing this in that instant we have this in the back of our minds remember if I take the derivative of I hat now DDT of I it is going to be perpendicular in that direction which is what the J hat direction and if I take the first derivative here if it's just turning these two I just turning see them as part of two vectors I can draw in another vector there it will have that as it's velocity vector so it's velocity vector this is that 90 degrees that's at pi over two radians it's going to point in the negative I hat direction so if I take the derivative DDT of of I hat what is that it's going to have a magnitude omega and it's going to point in the direction and if I take DDT of J hat it's going to be negative a magnitude and a direction that is fundamental if I take the derivative of this a rotating frame because this is going to rotate all the time or be on a path and we're talking just a magnitude of one so it's derivative it's also just going to be a magnitude of one a magnitude of one in that direction magnitude of one in that direction but it's moving at an angular velocity so it's one times omega J negative one times omega I hat that is so fundamental just to have this in the back of your mind because we start taking derivatives of this with respect to the world view this is going to be very important for us to do clear the blackboard I'm going to keep this on though because this is fundamentally important I'm going to take the first derivative of this get to an equation the second derivative of that get to acceleration and it's beautiful good there we go I've got to take it's first derivative because I want to know what velocity is and I remember that let's just write out something what is R of the intermediate frame with respect to the world view now these are parameterized curves parameterized functions vector functions so imagine I have some magnitude in the I head direction and some magnitude in the J head direction and I add them and that is my vector now imagine I'm just going to suggest that we have the X of point I in the I head direction and we're going to have Y point I in the J head direction so remember point I was the origin of our intermediate frame and this is in parameterized form so X is a function of T Y is a function of T of that point I as far as the world view was concerned as far as this origin was concerned and if I look at the position vector of point P in the I world in my intermediate world and I see point P moving in that direction that's going to be another function I'm going to call it X sub P of T I should just put the function of that that was in the little I head direction and Y of P of T there in the J head direction so those were my two if I just see them totally separately if I just looked at point I point I is moving remember that's the origin of my intermediate frame in the world view it is moving I can parameterize it so it is a point that is moving on a curve as a function of X and T and a function of Y of T and just to make a difference I do that sub script I and sub script P if I'm sitting on that rotating intermediate frame I see point P moving quite differently I see it as another function of an X of T I'm going to give it sub script P to differentiate it from that but it's also with respect to small I small J head two completely different things now if I wanted to know what the velocity vector is of point P in the world view that is the first derivative with respect to time I'm expressing these things with respect to time so I'm putting the dot there properly I should probably if it's not with respect to time say prime but let's put the dot there of of I with respect to the world plus the first derivative of P in the intermediate frame so let's do that let's take the first derivative of this one and then a second the first derivative of this one okay so velocity of point P in the world view what is the derivative of this with respect to time or it's first derivative please you're going to see me put a dot it should actually be R prime in R prime there because it's not always with respect to time but for this example I'm going to put a dot there it's my first derivative with respect to time DDT DDT instead of a prime there with respect to time not a prime but a dot okay so I can take the first derivative of this the first derivative of that that's very simple to do that is X dot of I in the I hat direction and Y dot first derivative of I in the J hat direction I can actually do the chain rule here I can also say then the first derivative of this but that's a constant that is actually a constant so it will be 0 in the first derivative of that one also be 0 show you now what I mean by that so that is going to be that that's the first derivative but now we have to take the first derivative of this with respect to the world view so I can't just say X prime of I hat and Y prime of P of J hat because this is 2 vectors that's a unit vector this is another vector and I have to do or a function of T and a vector so this is a product to rule a product to rule so I have to take the first derivative of this plus this times the first derivative of that look at it so I can say X prime of P in the I hat direction plus X of P DDT of I hat that is not a constant this thing is moving in the world view so I'm taking the first derivative with respect to the world view here this thing is not constant that one is constant this one is not constant so you've got to use the product rule here plus Y of P in the J hat direction plus Y of P just normally DDT of J hat got to remember to do that now this is very simple this is just the velocity so the velocity of P in the world view what is this this is just the velocity of I in the world view sorry of I in the world view it's first derivative this is that and if I look at these two these two terms I hope you can see that that is just velocity of P in the intermediate world because it's I hat and J hat and it's first derivative and first derivative that is just velocity first derivative of position but now I'm left with these two terms so I'm left with this plus X of P what is DDT in the I world it's omega J omega J hat and I'm going to have a negative Y sub P omega I hat because DDT of J hat remember we said DDT of J hat is negative omega I hat okay now this was simple what is this now you know it's omega cross R don't you but how do you get from here to omega dot R I'm going to show you if omega equals omega K hat and R now remember this is which R we'll talk about this omega I hope I don't forget but we've got to talk about R first this is R of P in the I world what is that that's X of point P in I hat and Y of P in the J hat okay if I say omega cross R of P in the I world that is the same as taking the determinant of the following it's the determinant of the following matrix I J K I always put a little negative there to remind myself that is positive negative positive if you know your matrix 0 0 omega X of P in 0 so if I take the determinant along the first row so first I close this and close that so it's 0 minus this so it's going to be negative Y P omega I hat and then along this it's 0 minus these two but there's a negative sign there so it's positive X of P omega J hat and for K it's going to be 0 so there's nothing there look at this that lot exactly the same thing so this is actually just nothing other than omega cross R of P in the I hat direction and there is our equation for velocity off point P as seen in the world view what you've got to be careful though is you have now there's the proper long way to write it out or I can just replace this with X of P omega J hat and negative Y of P omega I hat you have expressed this with capital I hats and lower case I hats okay you've got both in them eventually you want to answer all in capital I hats you have to go a step further and replace these with capital I hats and capital J hats and that we remember we had the cosine matrix where we could express one coordinate system within another coordinate system a rotated frame within a frame so we still got to do that step as well so we are combining something in the capital I hat and J hat world view with something in the lower case the only other thing we have to remember is omega what is that omega in the world view or in the intermediate view well it depends if you have something that is rotating at omega sub 1 and on top of that you have something else rotating on top of that rotating thing at omega 2 that omega 2 it's total is omega 1 plus omega 2 so it's all the omegas combined it's just this so this could just have well been omega sub 1 plus omega sub 2 I hope you can see in the corner there in the K hat direction so it's all your omegas combined those problems where you have this arms swinging around and the America around hanging there it is also turning so it is the angular velocity this whole thing has angular velocity around the world view then you just add all of those omegas so just be circumspect about this omega but certainly this is small this is lower case i hat j hat lower case i hat j hat capital i hat j hat so we still have to work on this before we are done eventually now for the acceleration but we really have to keep our wits about us so what you see there is my v of point p in the world view I've written it out in component form I just want to remind you it's nothing other than seeing the velocity of the intermediate reference frames point of origin with respect to the world view plus the point p with respect to the i view plus omega across r all the omegas added across r of p in the i view now depending on what textbook you have if it wasn't seen as world view world view i p sometimes it was seen as o a b then it's the velocity of point b is the velocity of point a plus v rel that is that little point moving relative to this moving frame and omega cross r so sometimes you see it in all different written in all different ways doesn't matter which way it was written as long as you understand where this is coming from so let's go with a component view and let's take the first derivative so we're looking at acceleration of point p in the world view so that will be the second derivative then of our position vector there of point p in the world view now we have to take the derivative in the world view of all of those components remember even if I did the product rule here it was going to be x double dot i hat plus x dot d dt of i hat but i hat is standing still it is a constant in direction and magnitude in the world view so the d dt of i hat is zero so those terms will just fall away even if you did the product rule there all you were going to be left with was x double dot of i in the i hat direction and y double dot or double prime of i in the j hat direction now you can see there I'm going to have a lot of terms there because you've got to apply the product rule and you've got to apply the product rule twice because you've got three terms x up omega and j remember how to do the product rule the product of three functions I'll show you now in case you forgot but let's do this one so the product rule for those two would be x double dot of p in the i hat direction plus what was d dt plus x of p prime what is d dt of i hat remember that's omega j hat omega j hat up there and we're also going to have then the same here we're going to have y double dot of p j hat plus y single dot of p omega i hat but remember that is a negative omega i hat and now I've got three terms there so what do I do first derivative that one that one first that one first derivative that one that one first derivative that one so I've got three terms that one for you let's have a look so it's going to be x dot of p the first derivative of that omega j hat now it's going to be x of p now the first derivative what's the first derivative with respect to time of angular velocity it's angular acceleration which is alpha alpha j hat and now the derivative of that that's going to be negative x of p and that is negative omega i hat there's already omega so it's going to be omega squared i hat omega square i hat and let's do the same for those I've got to get three components because there's three there so it's going to be negative i p omega i hat negative y of p alpha i hat and negative y of p omega squared j hat I hope I didn't make a mistake there scream out that I'm going to hear you if you see a mistake there so it's x of p x of p x of p y sub p y sub p y sub p then the first one gets the derivative then the second one gets the derivative and then the third one gets the derivative and the derivative of the d dt of j hat is negative omega i hat and there's already omega in there so it becomes omega squared and here it's going to be this one's first derivative then there's the second derivative and the derivative of i hat d dt of i hat is omega j hat so that becomes an omega squared so this very first one is quite simple that is going to be a acceleration of point i in the world view okay so that one is simple so we can cross these two out there's nothing spectacular there and I see something very nice here as well that would be a rel a sub rel in some textbooks those two so it's double prime there double prime there so that's going to be acceleration of point p in the i sometimes written as a rel relative so the acceleration of point p as if you're just watching from there to rotating frame but now we've got some other things to deal with I tell you what we do I see an omega squared there and I see an omega squared there so let's take them out as negative omega squared and what am I left with I'm left with something very cute x of p in the i hat direction and y of p in the j hat direction what was that that was nothing other than that was nothing other than r r of p in the i world the position vector of point p in the intermediate frame that's what I have there where did I where were they now here we go I've taken the omega squared out there and I've taken the omega squared out there and I'm just left with negative I took out negative omega squared x of p in the small i hat direction that was nothing than the position vector of p in the intermediate frame now I see I also have an alpha there I have an alpha there and where's my other alpha there is my other alpha there remember what is alpha alpha is going to be alpha in the k hat direction again acceleration in this direction vector pointing at you and what is what is there with alpha if we look at alpha there and we look at alpha there it's this point x of p in the j hat direction and we have a y of p there so what if we were to say what was r of point p in the i direction well that was x of point p in the i hat direction and y of p in the j hat direction and as before what would happen if we take alpha cross r of p in the i direction that would be the same as taking the determinant along the first row of this zero zero alpha and x of p y of p zero if I did that the determinant of this across this row what was I going to get I was going to get exactly this this one and this one remember how we did omega cross r now we're going to have alpha cross r in exactly the same thing so here we could add alpha cross r of p in the intermediate flame so we've got that one done so my two alphas are gone and all I'm left with is two of these things look there's a x dot omega j and an x dot omega j and a negative y dot omega j and a negative y dot omega j so what I'm actually left with here is two times x dot p omega j and I'm left with negative two times y dot of p i hat that's what I'm left with and I bet you now if you were to go and you were to take two omega twice omega you crossed it with r but it's not actually r it's actually v of p in the i direction if you were to do that you were going to get exactly that why because you would have written i hat negative j hat k hat zero is zero two omega and what was this this was the r of p in the i direction hat what am I doing that have been x dot of p in the i direction and y dot p in the i direction and zero and if you got the determinant of that along the first you were going to end up with exactly this so what do we add with in the end another plus two times omega cross r dot or let's put it as v of p in the i direction again remember you add all the omegas and there is my equation for acceleration of p in the world view there we go as you see it in the text books as I say sometimes you'll just see a sub a sub rel relative negative omega of r of p in the i direction so again I've got stuff in capital i and j and stuff in lower case i and j and I still have to change that lower case i and j's to uppercase i and j's by method that we'll look at but we actually know already so there is your long equation for doing that and it's simply taking the derivatives of the components and remembering that those little i hats and j hats they are not constant so you have to apply the product to quite simple to get to