A Nonlinear, 6 DOF Dynamic Model of an Aircraft: the Research Civil Aircraft Model (RCAM)

hello everyone and welcome to another video today I'd like to talk about developing a nonlinear six degree of freedom dynamic model of an aircraft now if you remember in some of our previous lectures we laid the groundwork for developing the flat earth equations of motion for a rigid body in space we discussed how these equations were actually valid for any a rigid body and the only thing that made them specific to a particular system were the external forces in moments and the mass and inertia matrix so in today's video I want to go over the equations and model for those parameters to develop a model of a twin-engine commercial transport aircraft in the next video we'll actually look at implementing this in a software package namely in MATLAB Simulink so the overall goal is that by the end of these next two videos you'll have an implemented model of an aircraft that we can use as a platform for future analysis such as trim linearization control design and a lot of other engineering studies so if that sounds like more fun than the seasons past a Disneyland let's go ahead and get started so the particular aircraft we'll be studying today is the research civil aircraft model or our camp for short so this is a theoretical model developed by the group for aeronautical research and technology in Europe or garter they published this excellent report entitled robust flight control design challenge problem formulation and manual the research civil aircraft model our cam the relevant citation is off there to the left I've also included the URL for the Garter web site in case you'd like to contact them to obtain a copy of the report for yourself so just to get a feel for what the vehicle is I've taken an excerpt from the manual that shows some of the physical parameters of the our cam model this is not modeled after any particular aircraft but these specs are actually very similar to a Boeing 757-200 so if you want to think about it that way you feel free to think in your head that this model that we're developing is very similar to this Boeing twin-engine aircraft note that the R cam document that I talked about earlier it's actually a very comprehensive document it covers all sorts of things obviously the plant modeling is what we're interested in but it also has things like sensor models actuator models control design and a lot more it's over a hundred pages of information but for today the scope of our discussion is really only modeling the plant I only want to understand how to develop a mathematical model for the rigidbody aircraft itself so really if you've got a hint a copy of the manual you're following along we're really looking at pages 6 through 19 primarily so the overall picture that goes along with our discussion today is we really want to develop the model this our camp aircraft model where you have inputs which are control surface deflections and coming out of the model is the estates of the aircraft now might actually be a really good time to start digging into the meat of the problem and talking about what makes this model our camp specific and the first thing that we might won't need to specify is what is the control vector for this aircraft so that's what's listed over here so in this particular model this aircraft has five control surface deflections the first three are pretty standard so you have an aileron as the first control input the second is the entire tail now it's not just an elevator the our cam model has a fully deflect above the entire horizontal stabilizer changes incident so keep in mind that that u2 deflection it's actually the entire angle of the tail it's not the angle of the elevator u3 is the deflection of the rudder and then since this is a twin-engine aircraft there are two independent throttles so u4 and u5 are throttle one and throttle two respectively now again getting back to our state vector again since we're modeling this as a flat earth rigid body object here are the nine states that we're going to be using so the first three are going to be our typical uvw which are the translational velocities in the body XY and z axes respectively the next three States four five and six are going to be our three rotational velocities which are going to be P Q and R which again we talked about earlier this is basically writing down the angular velocity vector expressed in the body axis and then finally the last three states or X seven eight nine are going to be our three Euler angles in the order of Bank pitch yaw so this is the picture that goes along with the overall theory of what we're trying to develop is building this model what we want to do today is we want to develop the equations that fill in that our camp box so in other words I want to replace that aircraft with a bunch of mathematical equations so I want to develop something that basically is going to consume you tell me what the current state of the vehicle is and you tell me what the current control surface deflections are and what we're going to be able to do is develop a set of equations that will tell you what is the state derivative or what is X dot so really that's all we're doing we're we're developing one set of equations here where I say X dot is equal to f of X U that's the goal for today's lecture to set the stage into kind of your appetite going what we're gonna do is once we have these equations done today in our very next lecture we'll think about replacing those equations with a simulant implementation so we can actually go around and fly this vehicle and simulate so that's our game plan so to execute that game plan we're actually going to take a series of ten sub steps so the first step that we're gonna do is we are going to talk about modeling some control limits and saturation on the control input vector then the next thing we can do is start thinking about some intermediate variables that we're gonna want to define and introduce to make our life a little bit easier downstream then getting into the meat of the problem step three is talking about modeling the aerodynamic force coefficients and the stability axis and once we have those we can then dimensionalize those to get aerodynamic force and rotate those to the body frame the fifth step is starting off with a similar set of operations for the moments so we're gonna start by looking at dimensionless aerodynamic moment coefficients about the aerodynamic Center in the body frame and then we can go ahead and and dimensionalize those to get actual moment about the aerodynamic center in the body frame moments are a little bit different in the sense that we're gonna need to take a step seven where we're gonna actually have to do a moment transfer to get that moment transferred from the aerodynamic center to the center of gravity of the vehicle and then you know that's actually most of the hard parts because the next step step eight is talking about we need to introduce some propulsion effects like things like what does the engine do to the vehicle in the dynamics and similarly in step nine we need to look at what does gravity do to the vehicle and then that gives us enough to basically place the model in explicit first order form that we're looking for so this is our game plan so what I want to do now let's jump over to the whiteboard and start working our way through each of these so we'll start with a discussion on how we're gonna model control limits and saturation alright so the first item to talk about is let's just right off the bat start talking about the limits on the control vector namely their saturation their upper and lower limits so again recall this is our control vector right there were five inputs to this system and aileron a fully positional horizontal stabilizer a rudder deflection and then two throttles okay so we need to now determine what are reasonable minimum and maximum values of each one of these channels okay so earlier I know I said we were only be going gonna be going up through page 19 but you know what you actually have to peak at page 20 of the RTM document to see values that they suggest in fact I'm gonna flash that an excerpt of that up right here on the screen this is the relevant verbiage from the our camp document where they specify what they think are reasonable limits for this model now note that they actually specify not just the absolute limits but they also talk about the rates that they suggest so in fact they're suggesting that you know your aileron your stabilizer your throttle all these things they're not able to instantaneously warm from one setting to another they take some time to slough or to transition from one to the other we're actually for our purposes to keep the model simple and to keep to minimize the number of states that are needed to represent the system we're gonna ignore the rate limits and instead we are only gonna pay attention to the upper and lower static limits so to speak so again those are the reasons that I have some of these sections circled in red so again those are the limits that we want to implement so again - with that in mind let's just go ahead and make a note that all of these need to be in the range maybe we'll do it well I'll kind of write it like such so the aileron can go between what do we have here - 25 and 25 degrees but again everything in this model is pretty much in Si so I need to actually multiply this entire vector by pi over 180 so again just make sure that you when you're actually implementing this or working with this we want to be using SI units so this is really - 25 - 25 degrees but it has to be represented in radians so I'm gonna take that into account here okay all right same thing the horizontal stabilisers so this is sort of asymmetric so this one it can pitch down and negative twenty-five degrees but only up to positive ten degrees okay the rudder can only go between or this one is symmetric and can go full negative thirty to thirty degrees on either side no the throttles are a little bit interesting they have this weird number of zero point five to ten and again they multiply by PI that they call this like radians or degrees however you want to think about it this number effectively is zero point five times 180 over pi sorry PI over 180 and same thing the upper limit is ten times pi over 180 again these are apparently somewhat arbitrary limits we're gonna see later on what these translate into in terms of maximum and minimum actual thrusts right because right now these are not actual thrusts right these are just plain old throttle deflections so they vary between these two numbers the only thing that's kind of interesting to talk about with this model here is that apparently you can't turn the throttles off here you can't make them go all the way to zero and again what we're gonna see later on down the road when we start modeling the engine effects we're gonna see that you can't actually shut the thrust off completely in this in this model so again these are somewhat arbitrary to think about here in these weird abstract numbers but if you just keep in mind that these are just a lower and an upper number we're gonna cut back to this when we start looking at the propulsion and the engine effects we're in step back down there and step 8 okay so again keep this in mind this is our our control saturation limits here so again we're gonna say that these deflections they can't go past these limits but in the model that we're building of the our cam we are gonna allow the throttle for example to instantaneously if the pilots started here at min throttle they could instantly ask for max throttle and the engine would spool up just like that it would instantaneously jump - max throttles right and same thing for all these LEDs these control surface deflections right we're basically saying that the control surface the actuators that are that are that are actuated these surfaces work infinitely fast okay so again we're trying to keep the models simple and it's simple we sacrifice some realism which i think is okay for our purposes so again I think this is a relevant discussion or a reasonable discussion for for our control limits so give me a second to pause the camera raise the bar and let's move on to step two all right so the next thing to think about and is intermediate variables what do I mean by that um to set the stage for that I think it's it's relevant to go back to where we are shoot what we're trying to accomplish Erik remember we're trying to accomplish this we would like to describe this function f here right where if you feed in the state at any given time and the control vector at any given time what this thing is going to do is it's going to be able to calculate the state dot right and it's going to do that using this function f okay now what's interesting about this is it's basically saying in this formulation what we want is we want all we want everything to be described the state derivative not to be able to be described as a function of just the states and the controls right so for example remember our state vector was like let's say X let's take X 1 for example X 1 dot right we said X 1 dot X 1 was you right it's the velocity in the body x axis so X 1 dot is u dot right so I need to find some expression for how you dot varies and as you can probably imagine this is probably some function maybe let's call it f1 right and this is probably a function of a lot of things right if you think about it it's probably a function of the lift that the aircraft experiences right the thrust you know the altitude you know all these other variables okay so we could write it down like this right if we knew how much the thrust affects you the acceleration and the body x-axis we could write this down similar with the lift all these other variables right if you think about it this form doesn't fit this format right here right because I don't need this function as a function of lift thrust all this other things I need it as a function of X and U so what we're gonna have to do is we're gonna have to make some substitutions right we're gonna have to do something and say that like Oh a lift is actually some other function maybe let's call it like a g1 the lift is a function of like the angle of attack and you know the dynamic pressure yada-yada-yada all these other things right and then we could substitute that in but we're still not there right because now what I would have is I would have a function of angle of attack Q throttle all this other things right so we're not quite there so I need to now make more substitutions like for example alpha right alpha is gonna show up probably a lot in our equation so we have to say what is alpha well alpha the angle of attack that is probably some other function I don't know how about like call it an h1 it's a function of maybe the U and the W right in fact we talked about the actual exact relationship and you know it's exactly a function of these two right there's a there's an eight and two involved with these two variables and this is actually golden because what's another name for you U is our first state what's another name for W this is our third state so here now we're okay because if we were to make all these back substitutions we finally get it to the level where you know what these states only show up here so the idea here is to make all of these substitutions in every single one of these variables until we're able to write this function f1 as only having X's use and obviously some constant variables showing up right so the idea with these intermediate variables is we don't want to actually have to explicitly write out all of these equations in one big long expression I'd like to to find some of these intermediate variables here right and these are functions of states and controls and then I can use the definition of like angle of attack or saengil a side slip or dynamic pressure any of these things I can use them later on down the road as long as I first define how they are a function of just the state's controls or constants right so with that concept in mind let's just write down a few of these easy ones so a couple these intermediate variables that might be helpful is first let's get the air speed okay so the air speed of the vehicle is you can probably imagine it just the two norm of the velocity well what I mean by that is right isn't that just square root of U squared plus V squared plus W squared Square rooted and again I think the better name for this is instead of calling this UV and W right I really wanted to say that what is U u is the first state X 1 what is V V is actually the second state x - 1 is w W is actually the third state X 3 so here this is a perfectly good definition because as you can see the airspeed now we have written down how the airspeed is related to only the states controls or inputs so great done with this this is the first one let's write this down so here's a good intermediate variable to use because airspeed as you can probably imagine is going to affect a lot of our equations of motion that we're going to be developing so right off the bat I think the first thing we should do is define how his airspeed related to the to the state vector okay let's do alpha right now the one we were just looking at angle of attack right this was uh it was basically tan inverse of what it was X 3 over X 1 right and actually this is sort of a slightly dangerous definition angle of attack right we talked about this in our other video when we were discussing the differences between a 1 an atan like this versus I for Quadrant a tan so I really wouldn't recommend doing it this way instead probably the better way to do this is probably to do an 8 and 2 of X 3 and X 1 and obviously this is assuming that a tan - you're using something like MATLAB where it expects the Y component and then the X component in that order again we talked a little bit about some of these nuances but again just make sure you read the manual on the a10 - that you're using and figure out which component do they expect as the first input argument and which component do they expect as a second input argument so in this case we're gonna go with this so here we go this is angle of attack all right that's a good definition for angle of attack okay let's also do beta angle of side slips so the angle of side slip that's the inverse sine of X - over the airspeed right and I'm gonna write it like this and this is actually totally fine because as you can see VA we already defined that as a function of all the states so really this expression is again just a function of the states controls or constants in this case just the states right so here we go so this is a good definition for side slip angle okay what else how about dynamic pressure let's call that capital Q here so capital Q is going to be one-half Rho VA squared okay so this is going to be dynamic pressure okay now this might warrant a couple of small bullet points with the dynamic pressure maybe let's note here so first off we're gonna use the notation capital Q is dynamic pressure vs. little Q which is going to be pitch rate okay so again just make sure that you're you're a little bit cognizant of that Capital cumulus dynamic pressure lowercase Q means pitch rate or I guess this is x one two three four five is it X 5 PQ yeah that more v stain okay the next thing that we should also notice is again we're trying to make these intermediate variables a function of just states controls or constants okay so let's look at this a half obviously that's a constant V a the air speed we already defined it up there we see that it's just a function of the states now this is where it gets interesting what is Rho for our model let's go ahead and consider this to be a constant 1.2 to 5 kilograms per meter cubed so this is basically your sea level air density right but as you all know right as you increase altitude the the air density goes down right so as altitude goes up air density goes down right at one point 2 to 5 it starts here than it does I don't know what it does I can't remember the actual table of how it looks like right but long story short the the density is a function of the altitude here right now the reason we are not going to try to model this is because if you think about this in order to model this curve in order to be able to handle this the fat that the air density changes with altitude we actually need to include the the altitude as one of the states we would need to be able to we would need to track how high the actual aircraft is in order to build this model here so this here obviously it requires H as a state right so we need to then implement what are known as the navigation equations to basically add three translational positions to our state to keep our model simple right now and to keep our model independent of location we're just going to go ahead and assume that this is a constant here so here is our our cam model assumption here's the real thing alright so again I'm just trying to point these out as we go through this I want to have a little small discussion on some of the sacrifices that we make in our model to make sure that it stays simple and easily manipulated but still captures I I think this still captures probably a reasonable amount of fidelity because you know as long as we're flying in a narrow band of altitude or somewhat near sea level I don't think we really need to worry about how the air density changes okay but again I think it's a good discussion to have it just be aware that we're making this this sacrifice and making this approximation and that we're incurring a little bit of error here okay so if we go ahead and do that if we accept that this is constant again we've succeeded in making it saying that this this intermediate variable Q is just 1/2 another constant times another intermediate variable which is just a function of the states so with these four I think we've got a fair bit of the intermediate variable set um just as a matter of convenience maybe what we will also do is let me let me erase this and so I think we hopefully have that written down by now we are also going to define let's define the angular velocity vector of the body with respect to the earth and this is expressed in the body frame okay so again if you remember this is just P Q and R right which in our case is X 4 X 5 X so this might be helpful later on down the road if we want to talk about vector operations you can obviously define this vector here which is again this is your angular velocity vector right expressed in the body frame okay and that's similar we can express the velocity of the center of mass of the vehicle expressed in the body frame this is going to be u VW right which is just going to be x1 x2 x3 right so this is translational velocity vector okay again Express in the body frame okay so that's great I think we're good here so with these sort of a six intermediate variables I think it's gonna lay some good frown good lay some good for a good framework for our subsequent discussion here on building up aerodynamic forces and moments building up propulsion and gravity effects etc etc okay so give me a second deposit board we'll move on a step deposit camera erasing board and move on to step three all right so let's move on to step number three which is modeling the non dimensional aerodynamic force coefficients in the stability frame now the reason I mentioned that it's in the stability frame is I actually want to bring up a screenshot here of section two point three point four from the our cam document where it says that the equation is that they're gonna give if you read that sort of last sentence right it says the reference frame for aerodynamic force and moments that is used in our canvas instability axis FS now a couple of things I want to note here first I'm all for this section for step number three we're only talking about the aerodynamic force coefficients later on we'll talk about the moment coefficients and I think it might be relevant now to talk about I think we need to draw a red line through that section where it says about moments because I'm gonna show you later that I think actually that they're actually presenting them in the body frame but that's a discussion for a couple of steps later so anyway let's just keep in mind now right now we're gonna develop some aerodynamic force coefficients in the stability axis okay so the first thing we should do here is let's look at how does the archive document define the coefficient of lift of the wing body combination so for the wing body combo the coefficient of lift for the wing body this is actually going to be a piecewise function where it's linear like this if alpha is less than or equal to basically 14 and a half degrees or that many radians otherwise it's actually a cubic function where it's a 3 alpha cube plus a 2 alpha squared whoops that's it this would be an a excuse me plus a 1 alpha plus a 0 otherwise okay so in this case a lot of these are constants so if you look up in the our camp document a lot of these things like this alpha lift equals zero this is gonna be negative 11 and a half degrees but again we're talking everything in radians and si so this is basically the Alpha at lift equals the zero this n right is going to be 5.5 okay and again if you look at what n is it's literally the lift curve slope so well tell you what I will draw a picture this and maybe that'll make a little bit more sense what this n is but it's mostly a number for now okay a 3 the our camp documents suggest this should be negative seven hundred sixty eight point five the a2 value should be it that they produce it as six hundred and nine point two times a one for whatever reason but they give you a value for a one oh no I'm sorry I just had a yes alright I was reading my notes sorry no it is just six hundred and nine point two the a1 value here is negative one hundred and fifty five point two and then the a zero term is going to be fifteen point two let's put a small asterisks on this we're going to come back to this number in a second okay so again I think it might be reasonable to let's just roughly sketch out what should this function look like and ask ourselves how reasonable is it what does it capture what does it neglect all of that kind of good stuff so here's see how the wing body versus alpha so we see that clearly there is something happening here at this switching angle of 14.5 times pi over 180 radians so again alpha here is in radians so at that point we see that if you're low if alpha is less than that you basically have a linear slope up to this point and then afterwards it switches over to that cubic function like such and actually here here's one issue that maybe I think we should address right away if you use actually these numbers the problem here is that right here at this switch there's actually a discontinuity so if you want to feel free to plot that out yourself you can actually just do it just plug these numbers in and you'll see that when you plug in 14.5 into here it doesn't equal the same number as 14.5 up here so there's this discontinuity as this discontinuity which is a little bit disconcerting later on down the road this is going to cause a problem so the issue right now is the R cam values basically leads to a discontinuity in CL Wing body ok so the easiest way to fix that here is I think is just go ahead and solve for I think we just need to bump up this this cubic Porsche nose here we need to bump it up so that it matches the linear portion so what that entails here is I solved it out myself you can do this yourself if you like but all you got to do is you got to add on a couple of decimal places to this and the prison that I'm sure maybe they just didn't go to this level of accuracy but what we need to do is in order to erase that asterisks I think you add need to make this a point two one two so I would suggest this modification to the our cam document and this if you use this value as the better you you will get rid of this discontinuity so let's go with that because I think this makes the picture a little bit more reasonable okay all right so this is what it looks like okay let's let's sketch out a couple of other points on this chart here so here's n right so here's the slope end this was our five point five okay and again what you can see right now is if you look at this long enough this letter n or this number n a five point five this is what this is actually DCL wing body D alpha in radians right so here's your stability derivative here is is positive five point five okay so that's the slope or your DCL wing body D alpha when measured in radians right okay how about your alpha when lift is equal to zero that's this number here so this here is negative eleven point five times pi over 180 I guess I didn't draw that very accurately this is this is not an accurate picture here but I think you understand what I'm trying to get at let's see what else is interesting oh here the critical value right up here where you get CL max this alpha crit right you could go ahead and easily solve that if you just go ahead and solve the derivative of this expression when it's equal to zero you'll actually get two solutions here because since this is a cubic by the time you take its derivative and solve for R its you get two values the ones that you get here or you'll get a two values so basically if you solve for roots let me just write this down solve for basically you take molla just I'll just say sulfur roots I think you know what I'm saying here right this is the yields two solutions here one of them is at twelve point three degrees the other one here is at basically eighteen degrees right so we clearly see this is not the one we want because this curve is supposed to extend down on that we're probably picking up this set this is the first root but that's not being used in our model so this is the root that we care about so what we're looking for here is to finish out our picture here is that the Alpha critical here that the maximum lift is going to be achieved at right around 18 I think it's 18 point zero one using my numbers times pi over 180 okay so that's the maximum of the critical angle of attack where you get a maximum CL of the wing body of above two point seven five one okay so that's what the wing body looks like let's see if there's things that we can we can discuss here and actually maybe let me see if I want to do this right now do ya where do they okay so again a couple of things to note here a couple of observations and things that we can think about now and try to get some physical intuition of what's going on so that later on when we actually fly the aircraft will understand its its limits and are sorry when we find the model will understand the limits so a couple things to note here first off is the CL of the wing body is only a function of alpha right so this is also this is probably clang in your head because you know that as you yaw the aircraft either nose left and nose right if there's some non-zero sideslip you're going to probably not get this efficient a lift curve here so that doesn't show up in this model and again I better put an asterisk on that statement that I just said here this is a true statement here but we're gonna see later on when we look at drag this this lack of dependence of our of beta it's actually partially due to the fact that we're looking at these in the stability axis and again I'm gonna have to punt and defer that discussion for about 10 minutes until we get there but let's just make a note of this here that if this will end up being true if we look at this we'll look at the full CL of the overall air craft and in our model it's only a function of alpha okay so right away that's one kind of glaring omission I think that is leads to a loss of fidelity of this aircraft a model right it's going to predict it's the same amount of lift even if your yacht out at 30 degrees as you would if you were flying you know straight into the wind so that's one thing to worry about okay what else is kind of interesting it does predict stall right so it has stall characteristics okay so if you look at this obviously right we see that at an at a critical angle if you increase past 18 degrees you're actually going to start losing lift so it does model stall so that's nice right so it's got that built into the model um however it only has the positive stall right it has it does not have a reverse stall meaning right reversed all right now what our model does right what this says is if we implemented like such as this just keeps going down and down and down and down down and down down down right to negative whatever negative in you so as you start decreasing the angle of attack of the aircraft I wish I had my model here but what should happen right is the lift should be the lift cohesion to be negative negative and then eventually just like you stall when you increase the angle of attack too far a real physical system if you increase if you decrease the angle of attack too far it should also have sort of a reverse stall effect right it should do something like this right so this reverse stall right is not modeled in our account in our model so it doesn't have that so again if you're flying the model and you're trying to operate in these weird extreme regions of the envelope like out here at negative 35 degrees angle of attack I don't think this model is going to behave or predict very well okay so that's something to think about okay again the other thing that we should should again note here and what this is going to be relevant later on this CL is in the stability frame right so really you almost want to save this a CL wing body and then I almost want to put like a superscript s on this to denote that this is in the stability axis because this is going to matter later on down the road so again keep in mind that this whole discussion here of the aerodynamic force coefficients all of these are presented in the stability axis ok let's see ok that seems pretty reasonable for the CL of the main wing body let me just double check to make sure there wasn't anything else interesting that I wanted to call out or discuss for this particular portion yeah yeah I think this is a pretty reason well ok tell you what give me a second to pause the camera and erase the board and let's talk about now how about the lift generated by the tail all right so the other major lifting body from this vehicle model is the lift from the tail so remember the tail in this model is a fully movable horizontal stabilizer and the first thing that we need to do to model this is let's look at the angle of attack of the tail so the model that the our cam proposes for the alpha of the tail is actually pretty darn interesting so it's the angle of attack of the overall aircraft but then as you probably know from your flight mechanics classes or from wind tunnel testing discussions that there's a there's downwash right from the main wing that is induced on the tail alright so again drawing your aircraft alright if you've got something like this you got your horizontal tail here you've got your main wing like such alright you've got some streamlines that depending on the amount of lift that you're generating right you've got this amount of effectively downwash that is generated from the main wing basically bending the streamlines so that there's a lowered angle of attack of the tail than the main wing right which you've got is here which is what this this epsilon term here so it does so this is effectively there you're modeling a little bit of the downwash of the tail okay so it's that plus a couple other things that we should mention again we talked about how the control right u2 was basically the Delta tail right so the entire tail goes up or down depending on what this u-tube values so actually you can just put you two right there all right so that and then here's a fascinating term that's added to their model here is plus they've got 1.3 x 5 x this parameter LT all over VA okay so let's break down each one of these and take a look at them so here again we said that the downwash here this downwash angle epsilon the model that they proposed is this here it's basically it's the epsilon D alpha times alpha minus alpha lift equals zero so again remember this was from the previous CL of the of the main wing body I think this was 11.5 times pi over 180 all right this the epsilon D alpha is just another constant that is given okay and the constant is 0.25 okay now if you look at this term again this is very interesting again remember we're trying to model the downwash here so let's examine this first so this epsilon is almost like the change in angle of attack at the tail due to the lift generated by the wing which you can kind of see here right this term in parentheses is basically the change in angle of attack from the from the alpha lift equals 0 so you can almost think of this as being proportional to the amount of lift is generating right because when alpha is at at when at alpha lift equals zero the May is generating zero lift right so therefore there should be zero down wash and as you start increasing the angle of attack of the main wing right you're generating more and more lift and this term right here this D epsilon D alpha is effectively telling you how much does that increase the lift or in this case the increases angle-of-attack contribute to the change in downwash so again this is a pretty reasonable model of a reasonable first order model of downwash so again this is a this is pretty great okay so that's that's downwash let's look at some of these other terms I think like we said earlier you two that makes sense right this is Delta tail and again since we have a fully moveable and deflect Abul tail yeah since you change the angle of the tail you directly change the angle of attack of the tail right that makes perfectly good sense now this last term right over here this this 1.3 times X 5 LT all over VA so again let's take a look at this this is basically if you look at this long enough this is actually including some dynamic pitch response the reason we say that here is because again X 5 right let's look at what was X 5 if you recall X 5 is pitch rate lower case Q right this L whoops sorry this is LT right this is just a constant it is the distance between it's effectively how far back the tail is so it's it's this distance here it's like LT is some measure of the distance between I believe it's the CG or maybe it's the AC I can't remember exactly what the station was but it's the dit it's the kind of the horizontal distance between the the tail so in the model here let me see what is the exact number here so the exact number for LT is 24.8 okay so this is interesting you can look at this and you can see that as this distance increases right the angle of attack change of the tail due to the pitching moment increases which makes sense right you're increasing the moment arms so as you pitch up and down faster you're actually changing the angle of attack again this whole term is proportional to the pitch rate so if you have zero pitch rate this term doesn't contribute at all but if your Q is nonzero the faster you pick up the more angle of attack change you see at the tail and again you also see it's inversely proportional to the airspeed right as the airspeed goes up this phenomena goes down which again it makes sense with this discussion of how you're basically looking at the moment arm or the distance between the the pitch point and the actual angle of the tail to see how the overall angle of attack of it changes so yeah this is the kind of an interesting portion that is added to this model so we should note here that this model for angle of attack of the scale it actually does include some dynamic pitch response if you contrast that with the seal of the wing body that we looked at earlier right seal the wing body was only a function of angle of attack of alpha right there was no dynamic response here so this is interesting that it actually captures this in the tail lift not the main wing body lift right so okay so now that we have the angle of attack of the tail the total CL of the tail okay the coefficient of lift of the tail is going to be three point one times s tail over s times alpha tail and here s tail this is the plan form area of the tail and the numerical value of that is 64 meters squared and then s is the platform area of the wing right and this is 260 meters squared okay so again this is a again maybe something that once I tell you what let's defer that discussion and we'll talk about where this s T and s came from in a second okay so now we've got the a we've got the CL of the main wing body we've got the seal of the tail so now let's go ahead and get an expression for the how about the total lift coefficient okay so the total lift coefficient in the arc ham model let's just call this CL it's going to be CL of the wing body plus CL of the tail and again be careful here we made an entire completely different video discussing manipulating these aerodynamic coefficients particularly this exact case where you're adding coefficients and there's a couple of gotchas of where this could go wrong so again here's the video if you haven't watched this please take a moment to check this out maybe now would be a great time to pause this video watch that other one so we can come back to this discussion of why you might need to worry about adding coefficients of different components together but well if you've already watched it I guess what we can say here is the nice thing is you'll see that the our cam designers they already took into account the potential difference between the normalization factors into account so you here's the s T over s so that was the exact problem that we saw in the other video where you have to be careful when you're adding coefficients they all have to be normalized according to the same reference area in reference length so luckily the our cam designers did that so we're allowed to just go ahead and add these two things straight together okay so um yes there we go here is a reasonable model for the CL of the entire aircraft so into this model it has baked into it all of these these features and all of these physical phenomena that we're trying to model things like you know the dynamic pitch rate where we're modeling the downwash at the tail we're modeling forward stall but not reverse stall and all these other kind of things that we talked about earlier okay so great that's total lift coefficient give me a second - now how about pause the camera raise support let's talk about total drag and side force all right so let's turn our attention now to the total drag coefficient now in the our cam model the drag model is actually pretty simple it's just given by this expression so zero point one three plus zero point zero seven times five point five alpha plus zero point six five four squared there you go so if you plot this guy up alpha versus CD you can see that it has a minimum here at 0.1 3 and that minimum occurs when baby basically solved the derivative this thing for alpha equal to 0 or sorry for you first solve this derivative equal to 0 for alpha and you basically get this alpha for minimum drag is sitting here at about minus 6 point 8 degrees and again let's do everything in radians because that's what this expression is okay so here's your minimum and otherwise it looks like this quadratic right something like that ok so this is what your CD versus alpha looks like there you go the thing that might be going clang in your head is let's make a quick note here um let's say at first glance right there is no beta dependence right so what this appears like on first blush if you look at it is right now it's basically saying that you're flying along you're flying along and if you are pitched down here at an angle attack of negative six point eight degrees you have minimum drag and as you increase or decrease the angle of attack in any other any direction your drag is going to increase yeah and then you're probably like yeah that makes perfectly good sense but you don't see there's no beta term whoopsies I just broke my airplane there's no beta term in this right and you might think well if you're flying along and you y'all out at some non-zero side slip angle you would expect the drag to increase as well and at first glance it doesn't appear to be in this model right I don't see beta is showing up anywhere well the thing that we should again emphasize is this little arrow up here right that little asterisks in the our camp documentation saying that this force coefficient is actually in the stability frame so in a second we're gonna see that this actually does develop some dependence on beta in fact this is exactly what you would expect though over here just for you know completeness sake over here is a couple of plots from some actual wind tunnel data where we took the model and we put it at a constant angle of attack and then we actually yot it so it does lists i as sort of the x axis but SIA is basically negative beta so anyway the x axis is your side slip angle in some way shape or form and as you can clearly see there's some dependence on sight some angle right as you change that side slip angle the drag goes up drastically right so that being said again this is one of these things I think we need to defer for just a couple more minutes and then we're gonna see that actually that dragged that dependence on beta on the drag actually does show up here in the model again the the picture we're showing over there from the actual wind tunnel data that is in the wind axis not the stability axis and this difference of the two frames is what's actually get causing this this I don't know I don't want to maybe say it's a confusion but it's causing this this this appearance of lacking this beta dependence so at this point I think I'm family I think I better move on to the next force coefficient so that we can actually get to formalizing that discussion I was just rambling over so the last force coefficient that we need to talk about let me drop down my airplane is let's talk about how about the side force so let's talk about the total side force coefficient okay so that one let's denote that with C Y right first side force and again this one is just gonna be negative 1.6 beta plus 0.2 for you three ok so again pretty interesting model again you three if you remember this is Delta rudder so as you would expect if you step on the rudder that is definitely going to influence the amount of side force it actually probably should influence the yawing moment even more but that's the discussion for a couple steps down the road if we're just looking at side force you see that by stepping on the rudder it does shove the aircraft over to one side or the other now here it does have beta dependence and again that's that that probably makes sense that this is explicitly showing up here because as you go off to the side you would expect the aircraft to be pushed over to one side or the other again we got to keep in mind that this isn't the stability frame so this is going to be something that we need to think about and in fact I think we're actually in a position to maybe think about that right now because we've defined all three of the force coefficients at this point we did lift drag inside force so the this is the our cam model of the four the non-dimensional aerodynamic force coefficients expressed in the stability frame okay so let's pause the camera array support and let's think about rotating this to the wind frame all right so let's think about actually how about if we rotate from FS right the stability frame which is what the model developed all these force coefficients in let's rotate from FS to the wind frame I don't know about you but I like to think about things in the wind frame it makes more sense to me I don't know if that's just you know I worked in a wind tunnel for three years and we always talked about things in the wind frame so I like things in the wind frame so let's go ahead and do that so the first thing I'm gonna do is let's go ahead and define some coefficient vector okay and this is in the stability frame so what I want to do here is I want to look at the I'm gonna put this in quotes the the force in the X stability axis right that's the first thing I'm going to put in there then I'm going to put the force in the Y stability axis and then when for the force in the Z stability axis okay so if you do that I think if you think about this long enough and again we talked a little bit about this in our discussion on wind tunnel testing if you haven't seen that video I please please take a moment to check that out first because we're going to leverage that information that we developed there so here the x axis I think everyone will agree that's actually negative CD right and again I better put this subscript s here because this is the CD that's in the stability frame the Y value this is actually just see why the side force in the stability frame and then the force that's in the z axis this is negative CL and the stability frame I think everyone will agree with that again and this is the reason I put this quote-unquote in force right because these are not actually forces right the coefficients that we just developed in the previous section of this video right where we said you know CL was CL of the wing body plus the tail and then the drag was this thing that had appeared to have no beta dependence and the side force was this thing where the rudder influenced and all that stuff that's what these three values are right that's what we developed okay and now I think to get a better understanding of what's going on it might be helpful now to look at this same force vector but in the wind frame right so what we can do is again I'm going to assume that you've watched our video on manipulating aerodynamic coefficients because what I want to manipulate right now is I want to take this vector of forces in the stability frame and I'm gonna rotate from the stability to the window frame and over the sites of angle right you remember this rotation matrix here see from the stability to the wind through beta this is a three by three that looks like what is it it's it's cosine beta sine beta zero negative sine beta cosine beta zero zero zero one right okay so what I can do is I can jam those the this in into here and rotate so again this thing here this CD this is this negative CD is what was our model this was the negative of the thing we just developed right at the zero point one three plus zero point zero seven times five point five alpha plus zero point six five four squared right it's the negative of that see why the thing we just developed this was negative one point six beta plus 0.2 four times you 3 and then the seal that was that ugly thing I'm just gonna write this a CL wing body plus CL of the tail this was a kind of complicated expression right but long story short right I can take all this go over to Mathematica and actually I can make all these inputs and all of these these substitutions get this vector and then rotate it through this to get the coefficients in in the wind frame all right so tell you what let's run over to Mathematica and do that real quick and see what we end up with all right so one of the first things that we can do is we can define input the Aero force coefficients in FS right so these were the models that we had developed on the board of CL C D and C Y and just to save you time I've already input those into the mathematical kernel and to find them elsewhere so if you look at them I've already plugged in all of the constants like alpha lift equals zero and all that kind of good stuff so these are the models that we end up with so what I want to do now is let's go ahead and put them in vector format so I'm gonna define what we called on the board I think we called it CF and the stability frame and maybe we we put a bar on top of this just to denote that this was a vector right so CF and the stability frame was going to be what do we say this negative C D CY and then negative C L all right that's what we got C FS / let's look at this in matrix form and we see it's just those three equip those three expressions stacked on top of each other and again we said some of the things out that maybe at first glance when clang in our head is particularly this dragged dependence we see that there's no dependence on beta in the drag okay however now if we go ahead and think about rotating this to f wind we are gonna go ahead and first let's define the window stability frame or so excuse me the stability frame to win frame rotation matrix and again just to avoid watching me input that I've prepared it off screen and here we go this is the rotation matrix that we defined and now what we can do is just go ahead and rotate that vector so I'm going to use this rotation matrix and operate on the stability frame coefficient vector and this should give us that same coefficient vector but in the wind frame so if I do that and then look at this in the matrix form here we go and now what we see is this first element here this is actually the drag in the wind frame and now you see that the beta dependence does indeed show up so for example if just to be perfectly clear we can print out all those three by just extracting each of these matrices each of these elements so here I'll just copy this here so the coefficient of drag in the wind frame it's actually it's the negative first element right here right that's the drag coefficient expressed in the wind frame this second element of the vector is the side force expressed in the wind frame that's what I'm printing out right here and again we just need to take the negative of this third element to get the lift in the wind frame so again just looking at that if I shift enter that here we go now you see that there's definitely a dependence on beta on the drag as we expected so in fact we could we could plot this if we wanted to let's go ahead and plot CD in the wind frame all right and let's maybe look at this when alpha is zero because you notice right there's alpha in here I think that's the only thing I need to to worry about yep okay so alpha zero let's plot this thing from beta how about I don't know minus 15 degrees to positive 15 degrees oh wait whoops hold on a second what did I get miss alpha beta oh sorry there's also a u3 here so there's a there's a rudder expression so I need to basically pick let's let's set the rudder to zero and there we go and here's our plot and maybe just to be explicit about this let's label the axes so the x-axis this is beta right in radians and then the y-axis is efficient of drag right but this is the coefficient of drag in the wind frame right and there we go this is kind of what we would expect and again this now matches up with what we showed earlier on the board when we were looking at wind tunnel data where the wind tunnel data showed a dependence on drag but that was obviously all expressed in the wind frame because it came from the wind tunnel all right and that's just how it decided to output that data how we we asked for it now what is also interesting though is if we come back here let's look at let's look at the lift term here's the lift in the wind frame and here we notice that still in the wind frame the our cam model does not have any dependence on beta so in other words this aircraft even in the wind frame it's it's still gonna produce the same amount of lift no matter what the side slip angle is so again something to keep in mind that we do model the drag dependence on beta but there is no beta dependence on lift in this particular alright so on to step four now we can go ahead and dimensionalize and get the actual aerodynamic force in FB so the first step of that is to actually get the aerodynamic force in the stability frame which is what we had just computed right so this this force vector in the stability frame is going to be negative drag positive side force and negative lift right again all of these were the values in the stability frame and then we were gonna get those is by taking the negative CD right the coefficient of drag that moment or sorry the coefficient of drag and the stability frame which was the model we had just talked about right which appeared to not have any dependence on beta but again that appearance was just due to the fact that we were looking at in the stability frame not the wind frame well now we actually this is the body frame so first steps first let's get this non dimensional value up to a dimensional drag so again all I got to do is just normalize that by dynamic pressure which again remember what spot was big Q not little Q and then times s right similarly the side force this was just our coefficient of side force times dynamic pressure times the wing area and then finally the lift in the the frame was the negative CL times Q times s so there you go this is going to basically give us the aerodynamic a force vector in the stability frame so again this is just the four the forces due to pure air dynamics right no engines no gravity no nothing right this is just air dynamics this is the vector expressing the stability frame now we can just go ahead and rotate to F be the body frame okay so remember that the discussion we just had we actually rotated this to the wind frame but now we actually want to go the other way I want to rotate this to the body frame so that's pretty easy the vector in the body frame is just related to the vector in the stability frame but I need to now rotate from the stability frame to the body frame through the angle of attack alpha or here right and if you remember this out our our rotation matrix from the stability to the body frame through the angle alpha was again our good old friend our three by three of cosine alpha zero minus a sine alpha zero one zero sine alpha zero cosine alpha there we go perfect and again if you need a little refresher on all these rotation matrices between stability wind body all that kind of good stuff check out our other video where we already discussed these aerodynamic angles alpha and beta and how to get these rotation matrices but if not if all you care about as a result here you go so at this point we are pretty darn good in terms of the aerodynamic forces expressing the body frame so that's step four let's let's move on to step 5 alright so let's move on to step number five where we want to model the non dimensional aerodynamic moment coefficients about the aerodynamic center right remember the the the moments are a little bit different than the forces in the sense of them we have to talk about which point are the moments acting about well we're gonna first talk about developing the moments about the aerodynamic center and now in FB now this is kind of interesting this is one of the first stops I want to talk about this part of the model now if you remember previously we flashed up that screenshots showing an excerpt of the our camp doctor where it claimed that all of the aerodynamics both the forces and the moments were expressed in in the stability frame I think that is a slight typo that I believe should be at least for the moments the moments should be in FB and I'm going to show you why later for now why don't you just take my word for it that really this is the moments expressed in the body frame so all these are in the body frame so maybe let's add a superscript B here so again this is where we're gonna deviate a little bit from what the our camp documents stated but you're gonna see that the equations that they laid out lead me to believe that this is actually expressed on the body frame let's just go with this for now and take a look at the model so again to save some time and avoid just me just watching you having you watch me just copy down what the our camp document has I've just written it up on the board ahead of time so again here we go we've got the roll pitch and yaw moment about the aerodynamic Center that's what the subscript a C stands for right all of these moments that express in the body frame are given by this expression here where you've got this ADA term this ADA term if we look at it it's a three by one vector and this is sort of your kind of typical static moment effects so you see that there's a rolling moment and that's induced by by being gone out at a different at nonzero sites live angle there's also here's your typical model here for longitudinal pitching moment this is a kind of the classical thing that you might see in an undergraduate flight mechanics class where they're saying you know what you in order to have longitudinal stability you better make sure that the slope of your pitching moment curve about the center of gravity has a negative value right or sorry that that pitching moment versus alpha has to have a negative slope you can kind of see that in the model where this is baked in right this if you think about it this term is basically pitching moment right and here's the dependence on alpha so you can almost see as the partial of if you change alpha how does this pitching moment change it changes by this coefficient right here there's negative three point one times this gobbledygook of all of these positive numbers so yeah on first blush just looking at this again keeping in mind that this is about the aerodynamic center not the center of gravity we'll get to that later but at first glance it appears that this aircraft is longitudinally stable because the CM alpha value is is negative here but again that's a little bit of a sidestep but I just wanted to point that out and then finally look at this last term this term is the yawing moment so this is basically showing that you've got some young moment dependence on both alpha and beta so if you're pitched if you're you have a non zero angle attack or a non zero side slip you've got some tendency to yaw and again depending on what the sine the si GN is of this derivative at a given location that's going to give us some inclination of the stability of the aircraft but again that's putting the the wagon R in front of the horse a little bit putting the card in front of the horse so let's just let's move on these captured in this eight of vector R kind of giving you some idea of the static stability I guess or the static behavior of the model and usually these are going to be the alot of the dominant terms are going to come from this a two vector now let's look at this next term here this one right here where I've kind of collected it this is slightly different than how the our camp talked to me and wrote it but I wanted to write it this way because I think it makes a little bit more sense in my head this thing here this DCM D X this is a little bit of abusive notation this is not a really the partial it's not a Jacobi a full Jacobian matrix because while we're doing here is if you look is this is really capturing the dependence of the pitching rolling pitching and yawing moment right as you just change Omega B with respect to en if you're this vector right here this was your P Q and R right so this is again interesting in the sense that the arcade model has a little bit of some dynamic behavior dynamic rate behavior baked into it so if you have a non zero rolling pitching or yawing rate right a p q and r the aircraft has a non zero angular velocity vector this three by three matrix which i've written right here is going to capture some of the coupling of how does that rate translate into pitching rolling or yawing moments that the aircraft is going to experience due to aerodynamics so this three by three matrix here is capturing a little bit of this dynamic rate behavior that's encompassed in the model so again that's why I gave it the name D partial C M with respect to not the entire state vector X but really just with respect to the state's four five and six which are P Q and R so again this since it's you it's not zero there's some interesting behavior of the model and there's some sensitivity of the models showing that if you have non zero angular velocities that will induce some type of moment on the aircraft which could either dampen or exacerbate those those angular rates right so again we're gonna have to do a little bit more analysis of this down the road but again that's not the purpose of this lecture today today all we want to do is just talk about building up a model here from that our cam document so here's the dependence of the the moments the aerodynamic moments on the state vector okay now this last term right here this is the effects of how the control surfaces are going to influence a moment so really this is how you're gonna be able to fly and influence the model here is through this this term more importantly it's through this three by three matrix here which I'll again given it this sort of abusive notation named it's not a full multi-dimensional Jacobian as you as this notation might imply but rather this thing right here is a three by three matrix down here which is almost like measuring your control effectiveness right because what this 3x3 matrix does is it takes your deflections right the aileron the tail and the rudder and this is gonna then transform those deflections into rolling pitching and yawing moments right because that's actually all those control surfaces are doing on an aircraft right you deflect a control surface you mean do some kind of rolling or pitching or yawing moment that's how you're gonna control the vehicle right well this matrix here is telling you how effective are those control surfaces and making that deflection and again if you just look at this it's got some interesting behavior some expected kind of results and some maybe that are a little bit unexpected maybe so like for example look at look at this two two element here right this two two element lines up with pitching moment due to the second control input which is the the tail so this is almost like your pitch effectiveness right how effective is deflecting the tail the horizontal tail or the elevator however you want to think about how effective is that at inducing a pitching moment so again you look at this term and you see there's things in here that kind of makes sense like like L tail the distance of the between the center of gravity and the tail right as you increase that you increase the moment arm of the tail so therefore it gets more efficient right same thing if you jack up the volume of the tail st right that are the size of the tail you make it a bigger you're gonna have a more effective tail so again that's a lot of this this stuff is baked in but you know by a very similar same logic for example like u3 let's look at you threes effectiveness u3 is the rudder you would expect the rudder to have some effect on the young moment here right that would be this element here the 3:3 element and in here if you look at this 3-3 element it's just a static number it doesn't have any by the exact same logic if you increase the moment arm right if you increase LT I would expect the young effectiveness of the rudder to go up but in the argument it's just the static number so it really isn't doesn't appear that that part seems to be a slight I don't want to say again I don't say oversight but it's it's sacrifice that we're gonna make to keep the models simple Oh and jeez I almost forgot I always seem to forget this I I this DC MDX matrix it's not complete you have to multiply this entire thing by C bar all over VA sorry X I want to make sure you did not miss anything else yeah there we go okay sea ice yeah don't forget that because that's going to going to matter okay in our model so uh yeah that's uh that's the that's the moment the aerodynamic moments one thing that we will maybe mention while we're standing here staring at this you'll notice here this is u1 u2 u3 but once you four and u5 right would you expect that u4 and u5 those are the thrusts right those should induce some kind of a rolling pitching and yawing moment on the aircraft as well right well we're gonna actually deal with those completely separately step five is just looking at just moments due to aerodynamics here so the engines and the propulsion effects on the vehicle we're gonna deal with that as a completely separate term here so all this is doing is just moments due to aerodynamics here and I guess it's more of a semantic discussion of where do you want to deflect and change or where do you want to draw the line between what is an aerodynamic effect like for example these control surface deflections are kind of aerodynamic right but we're gonna say that the engines are not aerodynamic so that that's fine we'll deal with that later okay so that being said let's move on to step 6 alright so on to step 6 we can now go ahead and dimensionalize that dimensionless coefficient to get the actual aerodynamic moment about the aerodynamic Center expressed in the body frame so again what we're gunning for is the actual aerodynamic moment about the aerodynamic Center expressed in the body frame yes I have enough subscripts and superscripts for you okay so this is actually winding ly simple so we just take the coefficient of moment about the aerodynamic Center alright that was the the vector that we just talked about and we said that again that was that vector that is it's already expressed in the body frame and now we just need to rescale this up now the nice thing about the our cam model is we can just scale this up by multiplying the entire thing times Q times the platform a times C bar so again if you watched our previous video on manipulating your dynamic coefficients you know that sometimes the rolling the rolling moment and the young moment are normalized by the reference span was the pitching moment is sometimes normalized by the mean air dynamic chord now lucky for us in this case the Archaean model decided to use the same normalizing length of c bar for all the moment coefficients so that allows us to rotate the vector slightly easier as well as it allows us to scale it back up and normalize this to an actual dimensional value of moments in terms of Newton meters by multiplying everything by the same scaling length here so that's really the only thing I think we should talk about here again this step six is ridiculously simple and it's simple because the our cam designers were nice and they thought about this head of time and normalized everything by the same number so when we come to implement this in code later on down the road in our preview in our following video this is going to be a basically a one-liner here okay so I think that's all we need to talk about let's let's now talk about how about maybe transferring this moment to a different location all right so onto step seven in step seven I actually left this up here and all a racist now I want to do this just to show you that this is really pretty simple what I want to do here is now instead of having the aerodynamic moment about the aerodynamic Center we want to now switch this and make this the air the moment about the center of gravity in FB okay so again here's the moment about the aerodynamic Center right that was that subscript okay we need to transfer that moment from the aerodynamic center to the center of gravity and again we discussed moment transfers already in our video discussing Windtunnel testing so I'm not going to read arrived that expression or explain in depth what moment transfers are if you need to review that please check out that prior video but what we need to do here is basically just perform a moment transfer so if you recall from the previous video in order to do this moment transfer all we need to know is basically the forces that are acting as well as the geometry of the distance between the aerodynamic Center and the center of gravity right the place that we're transferring it so to do that we can now then get the aerodynamic moment now about the new location the CG expressing the body frame okay this is going to be the aerodynamic moment about the aerodynamic Center expressing the body frame right but now the moment transfer part becomes I need to now add on the forces and luckily we already have the forces right those are the air the aerodynamic forces express in the body frame all right this crossed with the difference in position vector so here the position of our CG expressed in the body frame minus the position vector of our AC expression the body frame so here's our moment transfer equation and in this case again we already have this this is already here we already computed this in one of our prior steps let me just make sure that we already give you the same number here this was this was from step four right and then this one right here was from step six alright and then we just need to know this geometry so luckily that's given to us and we know this from the our cam document here so our CG is ticketed to toe where has this it's going to be okay so in the our cam document they give this they lead to specify a value of XC g which is sort of the x position or the the distance from some station so here zc g so they have some station where they're measuring all these distances to so the X distance of Y distance and the Z distance in that frame and the that's aligned with the body frame is its this value its 0.23 times C bar and then you got zero so the CG is right in line the centerline of the vehicle which makes sense and then zero point one C bar so it's close to kind of the waterline the zero waterline here but it's not exact here okay so that's the position of the CG and then the position of the aerodynamic Center is again they also define you know an X ay C of Y AC and a Z AC and these are just going to be zero point one two C bar and then a zero and a zero so great I think that gives you enough to basically uh define everything and we now are able to transfer the moments from the dynamic center to the center of gravity ok now this might be a good spot to leave it because I should maybe make a quick note here so some notes if you're following along with the our cam document so the our camp document has a typo at least I believe it's got a typo in the sign of this moment transfer so I will I'm gonna trust the expression that we derived in our moment transfer in our wind tunnel lecture we were talking about moment transfer so I'm going to use this expression in my software my implementation so just be careful that the at least a 1997 version of the document in my opinion has a typo I don't know if they corrected this in future documents and also this is I think where it lays bare and shows that this thing that we that the our camp document had laid out the aerodynamic centers the sorry the moments the aerodynamic moments about the aerodynamic Center I think it's very clear from this expression at least the and and the way they have it in the written in the document that these are actually expressed in the body frame because if you look at their document they have they have an expression which is very similar to what we derived here I'll be it with a small typo but there's no rotations between the stability frame to the body frame right they just say the aerodynamic Center is about the center of gravity expressing the body frame are something that looks very similar to this so that being said I think that that that's kind of the smoking gun to show that these the everything that they were talking about the the the things that we developed in steps five and six and now at seven all of those are talking about actually aerodynamic moments in the body frame okay so the thing that we need to worry about here at least we should make a note out of is that the our cam outlines forces or we should be even more careful we should say Aero forces aerodynamic forces in the stability frame okay but the our cam document outlines aerodynamic moments in the body frame okay so that's the I think the things that we should be be cognizant of but otherwise yes I think I think we've got it so at this point now through step seven we've actually fully characterized the aerodynamic forces and the aerodynamic moments and we've rotated all of them to be expressed in the body frame and for the moment case where we actually care about the point of the moments we've transferred them to the center of gravity so we're actually in a real good spot so give me a second to pause the camera raise the board and we'll move on to step 8 all right moving on to step number 8 where we are now going to quantify the propulsion effects so the propulsion effects here is again this vehicle has two engines and let's say that the force produced by the ithe engine it's just going to be the throttle setting times mg that's the our cam model version of the thrust here so for the first engine right this is just you four times mg right because you four was the first throttle input and again the thrust produced by the second engine is just you five times mg okay now if you recall right we said that u4 and u5 that they had to be less than or equal to I think what was the value the maximum value that we set out in step one I think it was like this weird 10° thing where again everything was converted to radians and again it kind of made weird kind of sense but not really in in a vacuum but now if we take this maximum number in conjunction with how the thrust is generated you can actually see that at max thrust so in other words where u4 and u5 are both equal to this maximum value right the total thrust that is basically f1 max right plus f2 max right and if you work out that number and in fact if you take that number and divide by the weight of the aircraft mg for the Archaea model you basically get around a thrust to weight ratio of about 0.35 here okay so interestingly again this is a little bit spooky but if you look at a 75 7 - 200 those things have they're equipped with a two of them and they actually they have pratt and whitney pw1000g around 42,000 pounds of thrust and if you multiply that by 2 to get the total thrust at the 7 for 7 with those two engines can sorry the 7 5 7 with those two engines and divide that by the max gross takeoff weight of the 757 of about 250 five thousand pounds you actually get a thrust to weight ratio of about 0.33 so again this number here in the our cam model is spooky ly close to what a seven five seven - 200 does here so again that's that's nice to know that this with these limits set up that the thrust is is definitely in the ballpark of reasonableness now this is just the gross thrust we're talking about here we actually need to get this in a vector right so we need to actually look at what's the engine forces due to the ice engine here and we need this expressed in the body frame okay so what we're gonna use here is let's assume that the the engines are perfectly aligned their thrust vectors are perfectly aligned with the body axis so that means that all of that force is going into the X component and there's nothing in the Y there's nothing of the Z so the engines are not you know towed in or out or angled up or down they are perfectly in line with the aircraft body x-axis here okay so this here is the force vector that one engine the engine I is going to produce depending on what the throttle setting is okay so obviously the total propulsive force expressed in the body frame it's just going to be F engine one right expressing the body plane plus F engine to express in the body frame right so great box this guy up because this basically gives us the total propulsive / I don't know if you want to call it an engine force right in FB right which is exactly one of the things we were looking for now in addition to both having the force we also need the moment right because probably these engines are not mounted exactly at the center of gravity in fact I'd virtually guarantee you that you know they're under slow line will each wing so there you crank up the forces or you crank up the throttle on these two engines and you're gonna get some kind of a moment so the moment the model here for the moment is going to be the moment due to the engines about the center of gravity of engine I oh my gosh we're even going even further into the subscripts expressed in the body frame okay this year we are gonna use our classic um it's going to be the moment arm let's call the moment arm mu I express in the body frame cross with the force vector which was which was up here right which was generated F I in the body frame okay so there you go that's what we need the only thing that's different here is that this mu I depends on if you're on the left engine or the right engine so here to kind of generalize that so let's say mu I all right this is a vector express in the body frame this is sort of the distance between the it's the moment arm between the CG and the application point of thrust of engine I so the way this is characterizing the Archaea model it's X CG minus X application point of thrust for an engine I and then it's Y application point of thrust of engine i minus y cg and then it's z cg minus the z application point of thrust of engine i so again all of these are constants that are going to be specified in the our cam model and that we are going to implement when we write this up as a MATLAB Simulink model okay but again keep in mind all these are just constant numbers so the the these guys right the the application point the thrust of these different engines are going to be different depending on which engine you're looking at the Y value for example is probably gonna be one's gonna be positive one's gonna be negative because you can have one on the left side of the aircraft one on the white right side of the aircraft I imagine these X's I believe those are the exact same value for both engines because they're both probably the same distance forward and then probably the Z application points of thrust are also similar because I can't imagine why you would not have those symmetrical for a vehicle okay so this gives us the moment from one engine from engine I so again the total maybe we should just draw this as a double arrow the thing we're looking for is the total engine moment about the CG expressing the body frame it's just going to be the sum of those two right it's just going to be the M ECG one expressing the body frame plus M II CG to express in the body frame so there you go so this one right here and we've got the propulsive engine force and FB and now right here is our model of the propulsive slash engine moment about CG in the body expressed in the body frame okay and again things that maybe notice here is we're gonna probably see that probably this z1 we would expect this moment arm to be nonzero because the engines are under slung on the vehicle most likely or I guess I guess depending on how the Archaea model is set up we're gonna be able to tell if they're under slung or on top of the lane I can't imagine they'd be on top of the way I've never seen a configuration like that but long story short we should see the effects that if this is nonzero when you crank up the the throttle right those two engines are gonna tend to have that either a pitch up or a pitch down I guess depending on the sign of this this term here so again interesting things also what's kind of cool about this is again having two separate engines and probably by having different y-values grossly different Y values depending if you're on on the left or the right side we're gonna be able to simulate things like single-engine outs and things like that we can look at asymmetric thrust settings so again this is kind of a cool extra tunable knob an extra feature of the model that we're gonna be able to exploit later on when we want to look at doing more engineering analysis okay all right I think that's enough said about the propulsive effects let's move on to step nine all right so the step nine here is talking about modeling gravity right that's the last sort of force that we did not take into account in our model yet so we need to come up with a model for the force due to gravity okay so the force due to gravity it's just gonna be the mass of the vehicle times the gravity vector okay so the question now becomes what is the gravity vector and how do we want to model it so the first thing we should do is maybe choose a frame to express this vector in and an easy frame to use right is our earth frame okay so in our earth frame or a vehicle here in Northeast down frame however you want to think about it if we are having if we're using a perfectly spherical model of the earth which is what we're gonna do for this our cam model this actually becomes really easy to express the gravity effects in this earth frame because the last time I checked right gravity this gravity vector there was no component in the X no component of the Y and basically it's just all straight down right at 9.81 m/s^2 last time I checked something as I drop it gravity pulled it straight down I didn't see it going in any other kind of direction right so this is a ridiculously easy model here right so here we go this is a nice easy frame to express the gravity vector in and therefore express this gravity force however what we need to do now is we actually need to rotate this to the body frame we need to rotate to FB right because we had the aerodynamic forces and moments in FB we had the engine from forces and FB now I need the gravity force and FB so again I need F gravity expressed in the body frame we have the force vector of gravity expressed in the earth frame so all we need to do right now is basically do our oiler angle rotation sequence to go from our earth frame or in this case the northeast down frame is basically the exact same thing right because we have a perfectly spherical earth model in the our cam so all I need to do is basically rotate from the earth to the body frame through my or other angles Phi Theta SCI and again we had a completely separate video discussing Euler angles and the Euler rotation sequence I'll leave a link for that in the description I think you probably should have seen this already if you're this far in our flight mechanics discussion so I'm not gonna bother writing this down I think you all know what it is I'm just gonna write out the result um and you see that the gravity vector it's a nine section it's a bunch of zero so that knocks out a whole bunch of stuff and makes this actually a pretty darn tractable calculation so if you write this whole thing out what you end up with is just negative G sine of X 8 which was theta right and then this one is G cosine X eight times sine of X 7 which was 5 p.m. okay and then we got G times cosine of X 8 and then cosine of X 7 all right now that we got to multiply this entire vector by the mass 2 in order to get the force so here we go here is the gravity force expressed in the body frame as a function only of states controls and constants here so again maybe one of the other things that maybe we should note here is that we're gonna assume mass as a constant so where we've got maybe an electric aircraft or something like that where you're not actually burning any fuel or the fuel is not changing in any appreciable rate here okay so there I think we've got everything we've we finally have done it we've characterized all the aerodynamic forces and moments all the aerodynamic are all they are the propulsive forces and moments and the gravity forces but maybe we should ask ourselves where is the moment about the CG due to gravity expressing the body frame right we had the aerodynamic moments we had the propulsive moments what about gravity well lucky for us I think you'll all recall that gravity acts through the center of gravity of the vehicle right so the moment arm due to gravity is zero right there's no moment due to gravity on this so this is a big fat zero so we don't need to worry about the moment that gravity is gonna cause on the vehicle Express in the body frame effect we don't need to worry about the moment in any frame right gravity is not gonna induce a moment on this object so I think we're good we've got all of our forces set up so let's go ahead and jump onto our last step alright so here we are at the last step where we are finally in a position to to assemble our explicit first order form equations of motion for our system so again I've gone ahead and written down our flat earth a motion for a rigid six degree of freedom model here or object so we see that again here's our translational equations basically this came from Newton the second law of translation and we see it here that this is the only term that was specific when I gets the master but mostly this FB right this was all of the external forces that were acting upon the body so in our case we saw that this was well there was gravity acting on the body and we just looked at in step nine how to characterize this vector well what else was there there was engine effects in the body and again we looked at how to deal with that in step eight and finally there were the aerodynamic forces expressed in the body and that's what we looked at in the pre and you know I can't remember exactly the numbers I think it was steps it was actually a lot of the first several steps it was almost like step two through gosh now I really want to see it steps yeah step three through seven here right kind of got us all these so these were where all the complications came into the problem right but we characterize we modeled all those using the our camp documents so we basically have fully characterized this set of equations for translational motion again the only other thing that's maybe worth bringing up here is is again the mass here the way we derived this flat earth equation of motion right is we made the assumption that the mass was constant right so maybe the only other slight side note that we should also talk about while we're discussing features of the Archaea model is we're assuming constant mass right okay okay great so done with this guy let's move down to what happened when we apply Newton's second law for rotation right that popped out the set of equations for our rigid six degree of freedom system right and we saw again the only thing that made this thing specific to a vehicle is this term right here so here we got the M CG we need all of the external moment about the center of gravity expressing the body frame well you know what that's it also what we just did in all of these steps right so here we had two things we had the engine about the CG expressing the body frame right this was how the two engines affected the moments and then we also had aerodynamics about the CG expressive the body frame again that was the complicated expressions that we had modeled in several of the intermediate steps and there was no gravity term like we talked about earlier so check mark on this we now have this equation fully characterized again the other thing that we should maybe notice here is maybe we should also again may make a similar note we're assuming IB is constant right so in these equations of motion we are assuming that this moment of inertia matrix IB which is definitely specific to the vehicle hole and non trivial that has to be constant so you don't have an f-14 or something that where you've got things moving around or wings sweeping back and forth this is just pretty much a static rigid body okay and then finally the last three the Euler angle equations right we already have these we didn't even need to talk about this because we already had an entire sequence actually two videos talking about how to track and how to relate rate information basically PQR or X four five and six how those relate to time rate a change of x seven eight nine or time rate of change of the Euler angles right you either use the Euler kinematic equations plus ons kinematic equations or quaternions to make our life easy and to write this down ah we're going to just going to use Euler's kinematical equations down here for the Euler angle state propagation so here we are and this is basically it we're done if you look at this long enough what's here on the left side on the left side we've got a big fat vector of X dot right because if you just stack these three on top one on top of the other you basically get the entire state vector does so all this is is you basically need to stack on these three equations so take this first set of a three by three stack it here then here's the second set of three equations stack it down here and finally our third equation which are or they're kinematical equations you stack that down here on the bottom and this whole thing if you look at it it's a big function let's call it f it's a function only of states controls and constants right all of this stuff that's what we spent the entire first part of the discussion building up is every one of these terms is now fully characterized and described only in terms of States or controls or constant things you know like these like lift slopes or other you know static numbers so at this point yes I know this was a long discussion and I had a lot to consume but we have basically fully characterized this set of equations for our arc am aircraft okay so just to keep the length of this video down I think this would be a good spot to stop the next thing we want to do is we are going to now run over to MATLAB Simulink and I want to be able to encapsulate this entire thing right this thing where now we're going to put in the state vector and the control vector and now we have the ability to calculate X dot right here's the equations X dot is equal to f of X U it's right there it's all that stuff we need to now implement this in a MATLAB script so I have a MATLAB function where I give it X and u it spits out X dot and that is the golden ticket that is the opening the floodgates to all this cool things we can do in terms of simulation engineering analysis and all that other good stuff so but again this is this is a whole video by itself I think that would take way too long if we try to jam this into the discussion tonight so would that be sin I think this is an excellent spot to leave it so I hope you enjoyed the video where we developed basically this entire nonlinear six degree of freedom rigid body aircraft model by basically following that our cam guidance with some slight modifications and I hope you've gained some understanding of some of the characteristics and the behavior embedded in this model so that when we perform some of this future analysis you won't be surprised and at least we can kind of idiot check ourselves to make sure that the results we get match our expectations as we understand the models so like I said I think it's a very place to leave it I hope you enjoyed the video and if so I also hope you will consider subscribing to the channel surprisingly if you just scroll down a little ways and click on that subscribe button it really does help me continue making these videos and I hope to catch you at a future discussion so until then I think I'll sign off for now talk to you later bye

Channel: Christopher Lum

Views: 14,389

Keywords: Flight simulator, aircraft model, RCAM, dynamic aircraft model, simulating flight

Id: bFFAL9lI2IQ

Length: 103min 32sec (6212 seconds)

Published: Sun Apr 26 2020